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Yorcut lvayosHrDepartment of Mathematics, college of General Education, osaka University, Toyonaka,Osaka 560. Japan

Ivfrrs.lHrxo TesrcucnrDepartment of Mathematics, Faculty of Science, Kyoto University, Sakyo-ku, Kyoto 606,Japan

ISBN 4-431-70088-9 Springer-Verlag Tokyo Berlin Heidelberg New YorkISBN 3-540-70088-9 Springer-Verlag Berlin Heidelberg New York TokyoISBN 0-387-70088-9 Springer-Verlag New York Berlin Heidelberg Tokyo

@ Springer-Verlag Tokyo 1992Printed in Hong KongThis work is subject to copyright. All rights are reserved, whether the whole or part of the materialis concerned, speci f ical ly the r ights of t ranslat ion, repr int ing, reuse of i l lustrat ions, reci tat ion,broadcasting, reproduction on microfilms or in other ways, and storage in data banks.The use of registered names, trademarks, etc. in this publication does not imply, even in the absenceof a speci f ic statement, that such names are exempt f rom the re levant protect ive laws andregulations and therefore free for general use.

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vlll Preface

g as finite branched covering surfaces of the Riemann sphere, and determinedthe number of parameters of Mo by the number of degrees of freedom of thebranch points.

In this book, we treat moduli spaces through Teichmiiller spaces andTeichmiiller modular groups as follows.

Let R be a closed Riemann surface of genus g, and let X be a marking onft, i.e., a canonical system of generators of a fundamental group of .R. Two pairs(R,D) and (B', D') arc defined to be equivalent if there exists a biholomorphicmapping f : R--- -R'such that /.(X) is equivalent to Dt. Denote by [E,X] theequivalence class of (R,E). Such an equivalence class [R, I] is called a ma"rkedclosed Riemann surface of genus g. The Teichmiiller space ?o of genus g consistsof all marked closed Riemann surfaces of genus g. It is verified that ?, has acanonical complex manifold structure, and it is a branched covering manifold ofthe moduli space Mn.Its covering transformation group is called the Teichmiillermodular group Modo which corresponds to the change of markings. It turns outthat Mn is identified with the quotient space TofModr, which has a normalcomplex analytic space structure.

The Teichmiiller space 4 h* appeared implicitly in the continuity argumentsof Felix Klein and Henri Poincar6, who studied Fuchsian groups and automor-phic functions from the 1880s. Robert Fricke, Werner Fenchel and Jakob Nielsenconstructed Tc k 2 2) as a real (69 - 6)-dimensional manifold. Fricke alsoasserted that ?, is a cell. Their method was based on the uniformization theo-rem of Riemann surfaces due to Klein, Poincar6, and Paul Koebe: every closedRiemann surface of genus S (> 2) is identified with the quotient space H f I ofthe upper half-plane .I/ by a Fuchsian group f which is isomorphic to a fun-damental group of .R. Then each point [R, I] in ?, corresponds to a canonicalsystem of generators of l- . Hence we see that [.R, X] is represented by a point inR6g-0 which is called the Fricke coordinates of lR,t). Moreover, the Poincar6metric on f1 induces the hyperbolic metric on .R, and the conformal structuredefined by this hyperbolic metric corresponds to the complex structure of .R.

One of Oswald Teichmiiller's great contributions to the moduli problem wasto recognize that it becomes more accessible if we consider not only conformalmappings but also quasiconformal mappings. A quasiconformal mapping meansa homeomorphism which satisfies the Beltrami equatiotr ut7 = pu". A Beltramicoefficient p measures the magnitude of deformation of a complex structure ora conformal structure. Around 1940 Teichmiiller discovered an intimate relationbetween extremal quasiconformal mappings and holomorphic quadratic differ-entials, and asserted thatTn is homeomorphic to R6g-0. He also introduced theTeichmiiller distance o\ Ts.

In the end of the 1950s, Lars V. Ahlfors and Lipman Bers developed thefundamentals of the theory of Teichmiiller spaces, and they gave rigorous proofsfor Teichmiiller's results. They also showed that To @ 2 Z) has a natural complexstructure of dimension 39 - 3, and can be embedded in A2(R) as a boundeddomain, where ,42(R) is the space of holomorphic quadratic differentials of aclosed Riemann surface E of genus g. From the Riemann-Roch theorem, it is

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XIaf,"Jard

Preface

those which determine a given point of "(E) is a Teichmiiller mapping. Then it

turns out that Q k > 2) is homeomorphic to the space Ar(F) _of holomorphicquadratic differentials on .R. Hence, ?s is homeomorphic to R6c-0. We also show

that "(.R) is complete with respect to the Teichmiiller distance.

In Chapter 6, using the Schwarzian derivative, we construct the Bers

embedding of "(R) into a bounded domain in ,42(.R.), the space of holomorphicquadratic differentials on ft*. Here, E* denotes the mirror image of .R. By the

Riemann-Roch theorem, Az(R-) is also identified with the (3g - 3)-dimensional

complex Euclidean space C3r-3. Using this embedding, we see that "(ft) has a

natural complex manifold structure of dimension 3c - 3. It is also proved that

the Teichmiiller modular group M odo is a discrete group of biholomorphic auto-

morphisms of ?r, and acts properly discontinuously on "0. This shows that the

moduli space Mo =Ts/Modc has a normal complex analytic space structure of

dimension 3C - 3.Chapter 7 treats the Weil-Petersson metric on 4. The holomorphic tangent

space of To at a point [.R,X] is identified with the dual space of ,42(R). Then

the Petersson scalar product on.42(R) induces the Weil-Petersson metric on ?n'

We give two proofs for the fundamental fact that the Weil-Petersson metric is

Kihlerian. Both of them a.re due to Ahlfors.In Chapter 8, we establish a beautiful formula due to S. Wolpert, which

states that the Weil-Petersson Kihler form on 4 h* a simple representation

with respect to Fenchel-Nielsen coordinates.We also give two appendixes. Appendix A deals with Schiffer's interior vari-

ation from the viewpoint of quasiconformal mappings. We explain Ahlfors' con-

struction of the complex structure for Ts, which was the first construction of its

natural complex structure. We also discuss variations with respect to degenera-

tions of Riemann surfaces. In Appendix B, we explain briefly the compactifica-

tion of moduli spaces.At the end of each chapter, there are bibliographical notes of books and

articles to which we referred in the text. The bibliography is not complete. There

is a vast literature relating to the theory of Teichmiiller spaces. We hope that

this list helps the reader to begin to explore these research papers. Any omissions

of references, or failure to attribute theorems, reflects only our ignorance.The authors are extremely grateful to Professor Osamu Takenouchi who rec-

ommended that we write this book. They also gratefully acknowledge the gen-

erous contributions of our friends and colleagues Makoto Masumoto, Hiromi

Ohtake, Hiroshige Shiga, a^nd Toshiyuki Sugawa, who read the original manu-

script, and made many helpful mathematical suggestions and improvements.

Yoichi ImayoshiMasohiko Taniguchi

October, 1989

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Chapter 5

Teichmffller Spaces

5.1 Analytic Construction of Teichmiiller Spaces5.2 Teichmiiller Mappings and Teichmiiller's Theorerms5.3 Proof of Teichmiiller's Uniqueness Theorem

Notes

Chapter 6

Complex Analytic Theory of Teichmiiller Spaces

6.1 Bers 'Embedding6.2 Invariance of Complex Structure of Teichmiiller Space6.3 Teichmiiller Modular Groups6.4 Royden's Theorems6.5 Classification of Teichmiiller Modular Transformations

Notes

Chapter 7

Weil-Petersson Metric

7.I Petersson Scalar Product and Bergman Projection7.2 Infinitesimal Theory of Teichmiiller Spaces7.3 Weil-Petersson I\{etric

Notes

Chapter 8

Fenchel-Nielsen Deformations and Weil-Petersson Metric

8.1 Fenchel-Nielsen Deformations8.2 A Variational Formula for Geodesic Length Functions8.3 Wolpert's Formula

Notes

Appendices

A Classical Variations on Riemann SurfacesNotes

B Compactification of the Moduli SpaceNotes

References

List of Symbols

Index

Contents

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179

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254

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arlr3 pue '6 snua3 ;o t; aceds rellnurqclal aql lcnrlsuoc aal 'raldeqc s-rql uI

f snuag Jo aJBdS JaIInurIr.raI

I ra+dBtlc

1. Teichmfiller Space of Genus g

ar-plane z a- Plane

F ig .1 .1 .

A coordinate neighborhood (U, z) of a Riemann surface .R is a pair of an openset [/ in ,R and a homeomorphism z of U into the complex plane such that forany element (Ui, ri) of a system of coordinate neighborhoods with U nU1 I $,the mapping

z o z i L : z i ( U n U ) - - - z ( U n U i )

is biholomorphic. This [/ is also called a coonlinale neighborhood of r?. Such ahomeomorphism z is said to be a local coordinale ot a local parameter on U of R.A coordinate neighborhood ([/, z) with p e U is called a coorilinate neighborhoodaround p, and z is called a local cooril inateor local parameter arounil p.

Local analysis on a Riemann surface ,R is reduced to analysis on domains inthe complex plane via local parameters. For example, a holomorphic funclionon ,R is a function / on l? such that f oz-L is holomorphic on z(U) for anycoordinate neighborhood (U,z) of ft. A mapping f of R into a Riemann surface,9is said to be a holomorphic mapping if wof oz-r is holomorphic for all coordinateneighborhoods (U, z) of R and (V, u) of S with /(U) C V. A biholomorphicmapping f : R --- S means a holomorphic mapping f of Ronto,S which has theholomorphic inverse mapping f

-1 : S - ft. Two Riemann surfaces l? a"nd S arebiholomorphically equiualenl if there exists a biholomorphic mapping between .Rand S. In this case, we regard ,? and ^S as the same Riemann surface and writeR = S. We say also that R and S have the same compler slruclure. Complexstructures, biholomorphic mappings, and biholomorphic equivalence may be andare actually often said to be confortnal straclures, conformal mappings, andconformal equiaalence, respectively (see $ 1.5).

Remark. A Riemann surface is a two.dimensional real-analytic manifold, and theCauchy-Riemann equation implies that local coordinates determine its orienta-

crqdroruoloqlq e sl U r- C : Jr leql ferrr e qcns uI araqds<n eql Jo ernl?nJ?s

xalduroc eql uro{ g ;o arrilcnrls xalduroa eq} eugap usc e^r 'g oluo araqds-rn

eql Jo dr ursrqdrouroeuoq e sacnpul ,n = (n)l uolltunJ eI{} eculs 'a.raqds-z aq1

ra^o Ar ereJrns Suualoc palaaqs-o^r} e 1aB aar uaqJ 'U 'I 'EU ul * ilI ( t'I srnx

rraql 3uo1e asr^rssore alsed pue O lo ttCI 'rg sardoc ollal a{eJ, 'aue1d-z aq} uo

lnc " sr {O < " I U > r} = ? ereqa'I - C- O ul"urop eq} otuo {0 >rrr,-I

I C > ^l = *H aurld-;1eq ra/(ol eql pus {0 < rnurl I C > t} = H aueld-}1eq

raddn aq1 qloq fllecrqdrotuoloqrq sderu .rn - z uollcunJ ctqdrouroloq aq;,

'z'T'ttd

'euo JeuJoJ eqt al"ts er* 'ala11 'uollsnulluoc ct1f,1eue fq ro ,,e1sed Pue ln)r'

Jo Poqlau aql fq Pelrnrlsuoc sr 1r 'dleotsselC 'panle^-e13uts sr "n

- (n)l - z

uorlcunJ crqdrouroloq eqlJo uorl)unJ asrelur aql qllqa uo e?"Jrns uueruelg aql sI

slql '4 - rn uorlrunJ cre.rqe3p eql Jo eeeJrns uuetuelll at{} aes sn lal (lxaN

'Qlt'{o} - ?) p"n (z'3) spooqroqq3rau eleurProo? o'n1 fq Peusep

sr C uo eJnlcnJls xaldurbc V'eceJJns uueurelg e osl€ sI'3 aueld xelduroc aql

go iorlecgrlcedtuoc lutod auo eql q qc$Iir{ '{-} n C = ? ataqds uuvu'ery aq;'(r'O) pooqroqq3rau eleulproo, auo fluo fq uarrrS sr O uo ernlcnrls xalduroc e'flaurep 'e?€Jrns uueruolg e st aueld-z xalduroa aqt ul O ul€ruoP ,traaa'1p;o 1sltg

sa"BJrnS uueurarlr 3o salduruxg 'Z'T'I

'[OO-V] re3ut.rdg Pus '[86-Y]

1e3ar5'[37-y] ueur.re3uts

pue seuof '[Ol-V] Suruung '[ag-V] ratsrog '[ga-V] €rN pue se{re.{ '[ZZ-V] ,tqoC'[gt-V] s.reg '[g-y] oIr€S pue sroJIqV 'ecuelsut roJ (llnsuol 'sace;tns uuerualg

;o ,{roeq1 1e.raue8 eql pue s}teJ esaq} .rog 'pa1e1n3u€Ir} aq u"c pue slas uado;o

sr$q elqelunot € seq e?eJrns uueuralg ,t.ra,ta 1eq1 u^{onl-llaa s-I tI 'uoll€luelro

slql qtla paddrnba s,tea,r1e sr eceJrns uusruarg € letll etunsse airr '.rageara11 'uor1

aueld-zeueld-or

saf,"Jrns uu?uraru'I'I

1. Teichmiiller Space of Genus g

mapping. This R is the Riemann surface of w = t/7. (See Ahlfors [A-4]' Chap.

8; Jones and Singerman [A-48], Chap. 4; and Springer [A-99], Chap. 1.)

Note that the Riemann surface R of the algebraic function w - 1/7 is also

regarded as the algebraic curve defined by the equation u,2 = z.

Finally, we see elliptic curves, i.e., tori from the viewpoint of algebraic curves.

For any complex number ) (# 0, 1), Iet .R be the algebraic curve defined by the

equation

w 2 = z ( z - 1 ) ( z - . \ ) . ( 1 . 1 )

In other words, .R consists of all points (z,w) e C x C satisfying algebraic

equation (1.1) and the point p- = (oo, oo). We can define the complex structure

of ,? by the complex structure of the z-sphere so that the projection r: E -

e, r(z,w) = z, is holomorphic. This r? is a two-sheeted branched covering

surface over the z-sphere with branch points 0, 1, I' and oo. The mapping

f : R - e, fQ,u) = w, is holomorphic. This function / is written as u, =

\rc=W] and R is a Riemann surface on which the algebraic function

u - {z(z _tG -, is single-valued.

The Riemann surface ,R defined by algebraic equation (1.1) is rega.rded topo-

logically as a surface illustrated in Fig. 1.5. Take two copies of the Riemann

,ph".", St, Sz with cuts between 0 and 1, and between,\ and m (fig' 1'3)'place them face to face (Fig. 1.4), and join along their cuts (Fig. 1.5). The

resulting surface is homeomorphic to the Riemann surface -R. Hence, R looks

like the surface of a doughnut. We call such a Riemann surface a torus. A torus

is also called an elliptic curue; Lhis name comes from the elliptic integral (see

$ 1 . 4 ) .

F ig .1 .3 .

'(od'a)rv

lo srolor?u?6 lo an1sfrs f)?ruouoc n t=f{ !g'!V} ro t=r1VAl'llV)} 1ec e16

r=f'(rlun aqr) r = r-[rB]r-tfvltrsltrvlL[ 6

uoll€ler l€lueu"PunJ eql sogsll€s Ptre 6g

t6, t"''rg Iy uro.r; pecnpul luql'luVJ '�"' '�llgr] '�lly] sess€lc fdolouroq eq1

fq pale.rauaE sr od lurod eseq q?lar U 1o (od'g1ttt dno.r3 leluaurePunJ eqtr'(f't ;St.f) seprs dy ql1,u uoE,tlod xaruoc e o1 crqdroruoeuoq ul€tuoP e 1aB ar*

uerll 'g'I 'EU ul se od lutod as€q I{lI^r ug'6V ' " ' 'r€l(I}t selrnf, Pasolc aldurts

3uo1e g, lnt pue d snuat Jo U ac"Jrns uueuell{ Pasolc e uo od lurod e e4e;'acottns uuourery uado ve Pe11eo

st a?"Jrns uueruarg lceduroc-uou y 'snua! allug Jo ac"Jlns uuetuerll pesolc € sI

a?€Jrns uu€r.uaru lceduroc /tJa^a l€qt u^rou{-lle^{ sl U'I snuaS;o $ snro} e Pue'6 snuaE;o sr araqds uuerualg aql 'f snua6 to ecottns uuDut?NA Pesop e PelIe?sl g'I '3lJ uI sB solPu"q d qtl,u alaqds e o1 crqdrouroeuoq ec€Jrns uueuary Y

sa?BJrns rruBtuaru pasolc '8'I'I

'e'r'tt.r

'r'r'ttJ

sef,"Jrns uu"uall|I'I

l. Teichmiiller Space of Genus g

Fig. 1.6. (g = 3)

Fig. 1.7. (g : 3)

L.L.4. Lattice Group Representations of Tori

We shall represent a torus as the quotierrt space C lf of the complex plane C by

a lattice group l-. since ur(z) = fiG4Q=U is a single-valued meromorphic

function on the torus R defined by equation (1.1), we can consider the complex

integral of Ilw(z) along paths on ft' For any point p = (z(p),u''(p)) on 'R, th9

elliptic integral @(p) is defined by selecting a branch of algebraic function u(z)

and a path from oo to z(p), and by setting

, / r ( t - l ) ( z - ) )

d zf 2 lP )o(d = J*

( 1 .2 )

saprs Eurr(;rluapr fq peul"lqo aceJJns e ss Pazllear sl J/C eceds luarlonb sq;'[z] sasselc ecuele,rrnba lle Jo st$suoc.7 ,{q C P JIC eceds luarlonb aq;

'e'r'ttJ

'z fq paluasardar sselc acuale,rrnbe aqf [z]fq alouaq '(z)L=

/ {lla J ) L lueluele ue slslxe eral{}JI J raPun Tualoar,nbaare C I ,z'z slurod. o/$1 1"t11 fes e6'a1ozau*rvut Iz =(z)L uollelslr"rl€

qlr^r pagtuapl sl J ) qlu+rvrn, =,L fra,ra'1ce; u1 '9 3o (9)ry dnorS ursrqd

-Jouroln€ arlrtpue aqt Jo dnodqns 3 se papreSar sl u JoJ ,7 dnorS e?I11"1 Y'U roJ ilno.r,6 acqyoy e J qcns II€c alA 'U

PIag raqunu I€eraql ra^o luepuadaput fpeaurl ere faql '0 < (z)Llr)L)u1 ,$sr1es zv pue rl spollad

aq? ecqs 'E, 1o pouad e Pall€? sI J Jo luauale fraag ';

Jo slueuela dq raqlo

qf,ee tuo+ rasrp qlrqAr senlel fueur .{1a1tugul seq (d)P uotlcun; eql t€q} ees a/tr

'{Z>u'*l'ou*rYut} = J3ur11a5'flarrtlcadsar

,k__4$_4 f or (u - z)(r - z)z f ofzp

- ,l

z=iv pue -rp ,l

,=tt

fq paluasarder are g'I '3rg ur Ig(Iy salrn? pesol? eldtuts aq1 Suop O Jo sanle eq;, 'd pue -d Eurutof q1ed e uo

spuedep 11 '{lanbrun peururelap lou s-I O 1e.r3a1ur atldqle slq} Jo enp^ eqtr

'U uo I ea.r3aP;o

g__,la_atzp

l€IlueraSlp crqdrouroloq aq1

3o'q1ed e 3uo1e'1er3a1ur auII € s€ (6'1) pleSal o1 alenbape arour sr II'tlroureg

'1er3a1ur Jo ql"d aq1 3uo1e uoll"nulluot cltfleue dq paunu

-ralep sr (z)np enle aql pue 9.1'31g ur -d lurod aq? o? spuodsarroe oo eJal{^r

se)"Jrns ur"urerll'I'I

8 l. Teichmriller SPace of Genus I

,4 with A' and B with 8' in the lattice of Fig. 1.8 by the translations 7r1,tr2,

respectively.Now, we define a complex structure of C / f . Let r : C + C / f be the projec-

tion, i.e., "(r) - lzl fot z € C. Introduce the quotient topology on C/f , which

is defined as follows: asubset U otC/f is open if the inverse image r-r(Lr) is

open in C. It is verified that C/f is a connected topological space.

For any two points [o],[6] € Cf l,we can take neighborhoods 7o,V6 of a,b

with r(I/") n r(%) - {. Since z is an open mapping, this shows that C/i-

is a Hausdorffspace. Moreover, for any point [c] e C/f , taking a sufficiently

small neighborhood vo of a, we see that n gives a homeomorphism of v" into

C/f . Let Uo = r(Vo) and zo: (Jo - Vo be a homeomorphism with zo(lzl) = z.

Then (t/",2o) gives a coordinate neighborhood around lalin C/f - Thus C/f

becomes a torus, i.e., a closed Riemann surface of genus I such that the projec-

tion zr: C --- Clf is holomorphic. The triple (C,r,C/f) gives an example of

universal coverings, considered in $2.1 of Chapter 2.

As is known in the theory of elliptic functions, the mapping [@]: r? '--' C lfsending a point p e R to a point [O(p)] e C/l- is biholomorphic. Hence we see

that a torus defined by equation (1.1) is represented by a Riemann surface c/lr

for a lattice group l-. In Chapter 2, we shall show that every torus is represented

by a lattice group l- in c (see the corollary to Theorem 2.13). conversely, it is

known that such a Riemann surface C /f is always biholomorphic to an elliptic

curve defined by algebraic equation (1.1). For details, we refer to Ahlfors [A-4],Chap.?; Clemens [A-21], Chap. 2; Jones and Singerman [A-48]' Chap' 3; Siegel

[A-98], Chap. 1; or Springer [A-99], Chap.l.

1.2. Teichmiiller Space of Genus 1

Let us construct the Teichmiiller space of genus 1.

L.2.1. The Moduli Space of Tori

we use the fact that every torus is represented by a Riemann surface c/f,

where ]- is a lattice group on c as in $1.4 (see the corollary to Theorem 2.13).

On performing the transformation z r* zf 4,if necessary' we may assume from

the beginning that the generatols ?r1 and 12 Lor I a,re the ca"nonical ones I and

r with Imr ) 0, respectively.Now, consider a lattice group

f " = { j = m * n r l m , n € Z } ,

where r € H = {r € C I Imr > 0}. As wa.s seen in $1.4, the lattice group I}

corresponds to a subgroup of ,Aul(C), and the Riemann surface R, = Cf l, is

a torus. Denote by r, the projection of c to c/f,. Notice thal cf f, has the

structure of an additive group.

'(z'dtsalH =tw'sl teqt

'(Z'Z)lSd fq g;o eceds

luarlonb eql qtl^\ PeUIluePt sl I,f41 }€tll sarTdurr I'I uraroeqJ'IrolJo sassBl? af,ual-earnba rrqdrotuoloqlq IIe Jo tas eqt ''a'r'r.uoq

to aeods ,Ppout eqt aq rW P"l'g aue1d11eq reddn aql

3o ursrqdrourolne crqdrouroloqlq e s\ (Z'7,)'IS1 ;l ,L fra.rrg 'ilno.r,6 Jolnpou eq1

( - lP+t'c ,''lI t = "9-po pue z) p'?'q'" l'7=

- Q)L | = (z'7,)rsd\ l?+!o )

dnorS aq1 1ec a,u 'aaog

tr 'l'(P + n)) - ([z])/ fq ua'rt3

sl ,U - ,'A I t Surddeu crqdroruoloqrq e uaql 'splotl (8'I) ;t 'f1asra,ruo3

'I = cq - pD s^eq a1ll

-lD + tcl '0 < (, ,nD;q _d - ,rwr

esuls 'I+ = ?q - pe l3q1 aas e^a ',! - (,t){or-/ pu"

f = (t[or-;f suotlela.r eqt urorJ (eJoureqlrr\{ 'srafielur eJ€ /p Pue ' ,?' ,9' ,D alaqlvr

,tP*'rP -"rQ * ,'t,o

1aB en 'r-! o, lueurnS.re etues eql 3ut{1ddy

'Ptt? - "9*te

'

ur€?qo ein 'a.ro;ereqa 'sre8alur ar€ P Pue '? 'q'D aleqlr

'P+tc=n=(1)l

'q+tP=1a-(,t')l

aleq a^{'ecua11 '? rapun 0 = (0U o1 luap,rrnba are (1)/

pue (,-r.)rf qloq snq1, '0= d ecueq pue'0 = (0)1 leql erunsse,teur eaa'.re,roaro141

'(9'6 eufrua1 lc) 0 + lc pue sreqlunu xelduoc 5rc ! pue ,lc ereIIA\'d + ra = (,)!

s€ ualllr^a sr / uaqa 'j t, o. 'crqdrouroloqlq q .f esn€ceg '(7'6 rue.roaql 15),!)Lo! -

lo"o-wrll qanf 3 - C,! Eurddeur crqdrouroloq e'sr 1eq1

'l P ! Wle

slsrxe eraql teql salldtq ureroeqf fuorPouoru aq1 'pelcauuoc t(1durs sr 1i acurg'ig oluo ,"A Io / Surddeu crqdrouroloqlq e q arerll l€t{} alunss? '1srtg

/oo"l4'

(e'r)

'I = cq - p,D qwn sta,aTut arD p PUD 'c 'q 'o e.taym

,Plrc _ tg*tp ,

uoxlnpt,

at17 filsr7os p puv t lt fi,1uo puD fi Tuapamba frllocttlilloutopqrq erD 'tg puo tf,

uol onl 'g auold-{1ot1 .r,ediln eql w / pao t squr,oil ony fiuo rol '11- tuaroaql

I snue5;o aoedg rall+urqrral 'Z'I

l0 1. Teichmiiller SPace of Genus g

It is known that the quotient space Hf PSL(2,2) is a Riemann surface (cf.

$2.4 of Chapter 2) and that a fundamental domain (cf. $a.2 of Chapter 2) for

PSL(2,2) is the shaded area in Fig. 1.9. Intuitively, we get the Riemann surface

H/PSL(2,2) bV identifying the sides of this fundamental domain under the

transformations z > z +1 and z e -lfz as is i l lustrated in Fig. 1.9. Hence

we see that the moduli space of tori is biholomorphic to the complex plane. For

more details, see, fot example, Ahlfors [A-4], Chap.7; and Jones and Singerman

[A-48], Chap. 6.

F ig .1 .9 .

Remark. A torus given by equation (1.1) depends on a complex parameter ,\(f

0, 1), which is denoted by ,Sr. It is well known that two such tori 51 and S1,

are biholomorphically equivalent if and only if there exists a linear fractional

transformation which takes the set of branch points { 0, 1, }, oo } of Sr to the set

of branch points {0, 1,^',m} of ,91, (see, for example, Clemens [A-21], Chap.

2.7). Thus we see that 51 and 51, are biholomorphically equivalent if and only

if )/ is equal to one of the following numbers:

1 ' � t l - 1. \ . + , 1 - ) ,, r r - ) ' ) '

Now, let G be a finite group of order 6, generated by cr()) = 1/) and

Sz(\) = I -.\ which are analytic automorphisms of D = C - {0, 1 }. This fact

shows that ML= D/G,where Df G means the quotient space of Dby G (cf. $2.aof Chapter 2). Moreover, we find a biholomorphic mapping F : D lG ---+ C, which

is defined uy r([.1]) = /(.\) with

( , \ 2 - ^ + l ) 3

l, \ - 1

/()) =t 2 ( ) - 1 )2

r ueaaleq a)uaraJrp aql l€rll reprsuo) uec e1t{ snql'sdno.r3 arr11e1 ueemlaq I* ,'J:/ rusrqdourosr eqt o1 spuodsauoc q?rq^,r'[((,r)tg)/] = ([(,r)tg])V pun

l((,4trtll = ([(,r)ty])V reqr qcns (od''A)ru * (od','g1rv: { ursrqd.rourosr

u€ sarnpur "A - ,"A:/ Surddeu ctqdloruoloqlq e^oqe aql leql ees e.tr etueg'{

[(,r),g] 'l(,t)tV)

] srole.reua3 go tua1s,{s le)ruou€) € seq qclqm (od' ,tA')r)L qtrw

pagrtuapr s1 ,1.(fpepurrS'.{1e,rr1cadsar'[(r)tg] pue [("r)ty] ot, pue I sPues

qclqlr\ acuapuodseilor aql repun (od'"A)to o1 crqdrouroq sl iI ueql'(odtrU)I,

Jo srol€reua3 go urals,ts letluoue? e a;rt3 ['g] Ptn [Jy] sasselc ddolouroq aqa'od

?urod es€q qtl^{'U uo (-r,)tg pue (z)ty sa,rrn, pesoll aldurts eulure}ap'.{1a.l.r1radsar'C ul r pue 0 uae^4,laq pue'I Pu" 0 uea^l}aq sluaru3as aql ''U

1o (od''g)rl dno.r3 leluetu"PunJ aql Jo lurod aseq € sB [0] - od a4ea'dno.r3

I"lueu"punJ eql Jo 1utod,r,ret.rr. eql uorJ luelua?els e^oqe eql rePlsuoc el!\

'01'T'tIJ

,,1

-

'(Ot't'3t.f aas) .{1a.,rr1cedsa.r',r. pu€ I o? (,r)/ pue (1)1

spuas (1)//z <i z uorleruroJsuerl eteurprooc a{} l€qt atoN '{q + ie'P * tc}

pue {l'IJ ''"'l(f

;o srolerauaS;o ecloq) eq} o} Sutpuodsa.r.roc auo eq} s€ pere-prsuoc $ ecuaragrp srql snIIJ, 'tJ = ,J Pue

'g + tD - (,t)t 'p + !? = (I)l

t€qt ees a^\'(g'I) uoltelar tuorg'(,-r.)/ pu€ (I)/ {q pelerauef ,.7 dnor3 e11161

eql ol ,,.7 dno.r3 ecrpel aq1 sdeu / ulnq.l'1'1 iuaroaql;oSoord eql ul se./ Joz(p + -ta) = (r)! Ull eql e{e} ''g J ,'A : / Surddeur crqd.rouoloqlq e rod 'd1aa11

-cadsar ','J/C-=,rgr pue 'JlC =,U Irol lualearnbe fllecrqdrouroloqlq luasaJ-dar qcrq,r.l /r pue r uae^ 1aq acuaresrp aq1 go Sutueaur er{} fpnls am 'ge;o 1s.tlg

I snuaC go acedg rallntuqcral 'Z'Z'I

I snuaD 1o aoedg rallnurq)ra; 'Z'I

+

12 l. Teichmriller SPace of Genus g

and r'corresponds also to the different choices of generators of ur1(R',po), i.e.,

{ F'(")1, [B'(")] ] and { / .(Fr(" ') l) ' /-([s'(" ') ]) ] (see Fig. 1.10)'Now, for any torus ft, take a canonical system of generators E, = { Fr]' [Br] ]

of the fundamental group r1(.R,,p) of R, and consider the pair (r?,Xo). Such

a Xo is called a marking on r?. Two markings Ep = {[1t]' [Bt]] and .Do' _={ [,4i], [Bi]] are said to be equiaalenl when there exists a continuous curve Co

ot R f.oln p to p' which induces the isomorphism Ts": T1(R,p) - t1(R,pr)

with [Ai] = Tc"(lAi) and [Bi] = Tc"(lBr)). Here, ?c'" sends an element [C] of

r{R,p) to an element [Co-1 . C 'C,] ol r{R,p'). For the definition of product

of curves, see $2.2 of Chapter 2. Next, two pairs (R, E) and (S, -Do) as above

arc equiaalent if and only if there exists a biholomorphic mapping h: S ' R

such that h. (D) = t . ( { lA ' , ,1 , [B ' t ] ] ) = {h-( [Ai ] ) ,h . ( [B i ] ) ] is equivalent to

Ep = 1[Ar],[Bt]]. Denote by [f i,X0] the equivalence class of (R,D). We call

such a lR, Do] a marked lorus. The Teichmiiller space T1 of genus 1 consists of

all marked tori.

Theorem L.2. For euery point r € H, let E(r) = { [,41(r)], [Br(t)]] be the

marking on R, - c/1, for which lA1(r)l and [Br(r)] conv,sponil to I and, r in

1,, rvspecliuely. Then [-R", X(r)] - lR,, , E(r')l in T1 if and only if r = r' .

Prool. Assume that [R",](r;1 -

fR,,,E(r')]. Then there is a biholomorphic

mapping h: R, , - R, such that h. (X(z ' ) ) = {h. ( [ .41(r ' ) ] ) ,h . ( [ ,B1(r ' ) ] ) ] is

equivalent to I(r) = { [41(r)], [Bt(t)]]. W" may assume that h([0]) = [0]by replacing h with hr(ltl) - n(lrl) - h(tO]) if necessary. Then, the definition

of equivalence of X(r) and h.(X(r')) implies that h.([.41(ti)]) = [,41(r)] and

h.([,B1(r')]) = [,B1(r)]. Take a l ift h of h with h(0) = 0. Then h() = az for some

complex number a. Hence we conclude that h(1) = d = 1, and h(rt) - ar' = r.

Therefore, we have r = T' , The converse is obvious. tr

Since every marked torus [ft,Xo] is represented by [.R',X(r)] for some r in

f/, this theorem shows that 7r is identified with I/'

Another method to mark tori is realized via orientation-preserving diffeo'

morphisms between tori instead of systems of generators of fundamental groups.

For that purpose, f ix a marking E = {Ft],[.B1]] on,R. Then any pair (S,/)

of a torus.g and an orientation-preserving diffeomorphism f : R- s defines a

ma.rking f.(21 = { /.([At]), / ([.B1]) ] on S.

Theorem L.3. Let R,S, anil St be tori, and let f : R -- S,g: R - S' be

orientalion-preserving diffeomorphisms. Then [S,/-(t)] = [S',9-(t)] in T1 if

and only if Sof-L: .9 --* S' is homotopic to a biholomorphic mapping h: S - St.

Proof. Suppose that [S,f.(t)] = [S',g.(t)].Take two points r,r ' € I/ for

which [,R, E]: lR,,x(r)] and [S,/.(tI = fR7,D(r')], where .R", R,', E(r),

and ':(z') are defined a^s above. Rega-rd / and g as diffeomorphisms of E" to

.R,,. We may assume that their lifts I and f send 0, l, and r to 0, 1, and r',

respectively. Thus we obtain a homotopy between f and' i by setting

'(Z'dlSa /(ff); qrp pagltuaPr sI uol;o l,;ig aeeds

qnpoutrr eql pu€'(U)J.to lc€ ol PelaPrsuoc s\ (Z'Z)IS1'acuepuodsalroc slql^A'(Z'Z)lSa I t ol Sutpuodsarroc (U)Z;o ursrqdrourolne erqdrouroloqlq aqlsI *[fr] uaql '[r1oo"t''A) = (l't ''U])-["] fq (U),, uo uoll?e EI eugeP Pu"!1as1r oluo ""A = A loursqdrouroaglP e sr qctq^r 'r-(("')L1o"'rt) =.'! tnd '(U)J

ui 1t"llr;"""rrti'!,Ul = l(')tS,(r)!'] ulelqo "^ r("t)Llo"tqo'! = (')r;'o'r1 ecurg'l(i)'ri = (Ir])'rl fq ua,rrE 'A <- Q)tU :'t1 Surddeu crqdrouroloqlq e sacnPulz(p+-tc) = (t)'! fq paugap (C)t"V;o'r1 luaurala aql'I{ 3 -r. 1utod,(ue ro;(prr€q raqlo aql uO 'lQ)L! .(t)fu] of [{']g] Surpuas eeuapuodsauo? eql,(q uarrr3,tpeturrd q (U),2 uo l, go uoltc€ eI{J '(S'I) urroJ eql w uol}"turoJsuerl l"uollc€rJr€aurl e sr. (z'?hsd ) ,L luaurale .{ra a 1eq} 11ecell

'(u)z u" spe (2'7,)ISddnorE .relnpou aql pq1 ureldxa e,$ (U)J p I7 uorlecgrtuePl aql Sursn 'ruo11

'arnlcnrls xalduroc e seq oEe (U).f ttqtlno surnl 11 'arogaraq; 'l't ''Al ol [(r)3'

''g] Surpuas acuapuodseuoc er1] raPun

(U)Z qU^ t7 fgluapl u?f, a/n 'g'1 uraroaql Pu" $lrsruar Sutpacard eql urordt(ef); fq pelouep q qcg,ra 'g

1o acods ren!,uqrre; eql '(/'5) sled qcns 1o [rf '5]

sasselc acualelrnba IIs Jo les eql ilet e1l 'rS * S :q tutdderu ctqdrouroloqtqe o1 rrdolouroq sl ,.g .- ,g : ,-lo6

't'l ruaroaq;, ul se 'JI fpo pue y. Tueyoatnbaere eloq" s (f ',S) pue (/'5') s.rred oil,1 ?eql fes ean'putut ut stql qfl1l

.(t)g = ((,r),<)'(,;) 1eB a,r,t '.re1ncr1red "l'l?)"!l = (lrl)"1Eurllas ,(q 'U - "2 : { ursrqd.rouroeJlP Sur,uasard-uollelualJo II? sacnPul

'c)z'6ffia=Q)'!

Surddeu reeutl eql teqlaas e al 'l(")S'oA) = log

'S] pue [(".r.)3' '"2f = k'U] rsqr qsns l? ) !'o! slurod

om1 3ur1e1 ',(1en1cy't(f)T'S] = [d3'g] ret{t q?ns S' +- A i;f urstqdrouroa;-;rp Suur.rasard-uor1e1ue1ro lre PuU "^ '[d3'5] snrol pe{reur f.rerltq.re rr€ roJ

tr '[(f)'t'rS] =l(S)'l '5i] teql saqdurr qcrq'u '1ue1e'unba ele

,g uo (g3)*f pu€ (3').(/"q) s3ull.reur t"qt ees et 'og.{q pacnpur ((a)0',g1rt,

* (("it)1or1'r,S)r:r : "c; rusrqdroruo$ aqt fg 't j ? ; 0'('d)tg fq uarr3 q qtlq/'a

('d)6 oq ('i\!rrt ruor; ,S' uo elrnc snonurluo? e eg. oC ?e'I'3 Euqreu aqt

rog lurod aseq e aq od 1e1 pue 'f pue /or1 uear$leq ,tdolouroq 3 3q (I i t i O)

,S * U : rf, p"I'ctdolouroq e.re S: ol U uror; f pue ;l'or1 sEutddeur o/'al 'V Eurddeur

crqd.rouroloq.rq € of crdolouroq sI rS * g : ,-to6 1eq1 esoddns 'd1asra,ruo3

'f1r1uapr eql ol ctdolouroq sr ,? - 'tg : r-to6(ecuag 'f ptre ;f uaenlaq rg fdolouroq e a^eq ear '[(z)rd'] = (lr))',t 3q?tnd

'r; I;0 'c) " '(r)0r+(z)!(?-t)=Q)'4

8II snueC 1o acedg rapurqslal 'Z'I

L4 1. Teichmfiller Space of Genus g

1.3. Teichmiiller Space of Genus g

In accordance with observations in the previous section, let us construct theTeichmiiller space of arbitrary genus g in two ways.

The first construction is given by considering marked Riemann surfaces. Asystem of canonical generators Dp = {lAil,lBil}f=t of a fundamental group

n{R,p) of a closed Riemann surface E is called a marking on E. Two markingsDp = {[Ai], lBil]f=1 and Dp,- {lAil,pf}fu on it are equiaalent if thereexists a continuous curve Co on .R such that [,ai] = Tc"(AiDand [Bj] =

fc"(Bi l ) for I - 1 , . . . ,9 , where ?s" is the isomorphism of zr1( .R,p) to r1(R,p ' )

sending any [C] to lC;r . C . C"). Let Do and Eo be markings on closed

Riemann surfaces i? and ,S of genus g, respectively. Two pairs (.R, Xo) and(S,E) are said to be equiaalenl if there exists a biholomorphic mapping

h; S - 8 such that the marking h.(E) = {h,(lA!i),h.(lBjl) }j=r it equiv-

alent to Ep = {lAil, lBil} j=r.Th" equivalence class of (R,Dr) is denoted by

lR, Ei and called a markeil closed Riemann surface of genus g. The Teichrn'iiller

spaceTo of genus g is the set of all marked closed Riemann surfaces of genus g.

The second construction is given by considering orientation-preserving dif-

feomorphisms. Fix a closed Riemann surface R of genus g. Consider an arbitrarypair (S, /) of a closed Riemann surface .9 and an orientation-preserving diffeo-

morphism f : R ---.9. Two pairs (S,/) and (S',g) are said tobe equiualentifgof-L:,S - S'is homotopic to a biholomorphic mapping h: S - S'. Let [S,/]be the equivalence class of (S, /). The set of all these equivalence classes [S, /]is denoted bv "(r?) and is called the Teichmiiller space of R.

As in the case of tori, we assert that the Teichmiiller space ?o of genus s(>- 2)

is identified with the Teichmiiller space ?(.R) of a closed Riemann surface -R of

genus 9.To see this, first fix a marking D = {[1i],lBil1]oi=t on .R with base point

po. Corresponding to a point [^9, /] in T(R), a marking /- (X) on ,S determines a

point [.9, f.(t)] in "r. It is noted that this point [.9, f.(D)) in ?, does not depend

on a representative of [S,/] in ?(.R), which is seen from Lemma 5.1. Hence we

define a mapping A2: T(R) --- To by setting

az(lS,/l) = [S, /.(t)]

for any [S,/] € T(R).

Theorem L.4. The mapping A 2 : T(R) ---+ To is bijectiue.

Proof (an outline). The injectivity of @s follows from the se'called Nielsen's

theorem (Ilarvey [A-41], p.43). It can also be proved by Lemma 5.1. However,

we shall give an intuitive explanation for its injectivity. Suppoee that two points

[ ,S, / ] , [ .9 ' , g ler@) sat is fy @.r( [S, f ] ) =OE([S' ,e] ) , i .e . , [S ' . f . ( t ) ] = [S ' ,e . (X) ]in Tn. Then we can take a biholomorphic mapping h of S' onto S and an

orientation-preserving homeomorphism go of S onto itself, which is homotopic to

the identity, so that gy - looho! coincides with / on each Ai and Bi $ f i S g).

ueql'[/3' 'S] = [(f)V'5r] 3ur,{;sr1es.g *- g:rf ursrqdrouroeluoq Surrrraserd-uorleluarJo ue slsrxe araql 1€rll /r\ou{ e^r 'a.roqe palels uaroeql s.uaslelN.{g'smo1o; s"e parrord eq osle uec 7'I {uaroeql ut dlt,rtlcafrns aqJ'tlrDuev

E '[f 'S] =l(5).{ '5'l acuaq pue'1ua1e,rtnbe are ,3 Pu€ (g)Y tnql

s,r,roqs / Jo uollrnJlsuol aql 'S - U : / usrqdroruoagrp Sur,trasard-uot1e1uat.roue o1 .{llerrdolotuoq zf ur.ro;ap tlualoaql Eurqloours aq1 ureSe Sursn .'tq ecuaq pu€'rng

lo pooqroqq3reu e ur qloorus dl.ressaeau lou sI z, slq;,'r11 uo 06 = 7'6 puern - A uo f - z6 WqI os S r- g : zf ursrqdrouoegp Surrrrasard-uoll€lualJo u€

augeq 'r1lg uo Id uorJ tp uo o6 usrqd.rouroagrp Eut,rreserd-uolleluelro ue lf,nrls-uoc '3[ ro; se fe,r,r atu€s eq? uI '{qp pesol? e o1 crqdrouroagp q Pue 72 sul€}uoc

qcrqaiodgo t4 pooqroqq3tau e ar1e1 'aroruraqlrng '(tD| - S * If -y :16 tustqd-rouoesrp Surarasard-uolleluelro ue o1 flectdoloruoq f ruro;ap '(6'1 ureroeql'g 'd*qC '[ZfV] qsqg) ueroeql Surqloorus e Sursn 'tnqtr 'lg p* fy qcea;o

pooqroqq3rau € uI qlootus fpessacau 1ou st f sHI'(r)t-S <- n-A td urstqd-Jouroeuroq Sutrrrese.rd-uoll€luelJo ue ulelqo e&' I) -A o1 g 3ut1celord 'alo11

'(g'g uraroaqJ, '8 'deqC '[Al-V] qsrlg) tV - "? t,! ursrqdrouroagrp

Surarasard-uorleluarJo ue o? spualxa sJ'$ql teql u^{oqs q }I'g -uyg uo tq1!^{ sepl)ulqx eJ' qcrqa roJ syg *- ive : e;l usrqd.rouoeglp e lcnrlsuoc ue1

e A ueql 'flu 5 ol Surpuodselroc UVQ uo 5i 1as eql raplsuo? pue '4srp pesol?

e o1 crqdrouroaglp sl qrlq^a U ut od;o n pooqroqq3tau 11eus {lluatcglns e a{"J.aV

lo uyg ,{repunoq eq} o1 a! p uor}crrlser aql aq ! pl 'l,S pue ' !,V ' lg ' lV

sdool pe Jo uorleluelro eql selresard 3/ 1"ql eurnss€ eir e.rag ' 6' " ' 'I - f 1e rol

{'d} - lS = (oit} - !g)21 '{od} - t,V = (od} - !y)s1

Surf;sr1es {odl - ,Coluo {0d} - C Jo 3;| usrqdrouoagrp Sur,traserd-uotleluelro ue a1e1 '1xap

'(,2'S) uor; fear

arues aq? ur peurclqo {srp lrun pesolc st{} "y fq elouaq 'uy {slP }Iun Pasol, eql

o1 crqdroruoaJlp sl d leqt asoddns feur arvr '4,;o xalra,r qcee punor€ ernlcnrls

alqerlueragrp Surceldag '(f 't '4.f ;c) aueld eql uo saprs f6 qll/'^ d uo3flod Pesolc

e pue oU ueearleq ursrqdrotuoagtp Sur,trase.rd-uorleluetro ue s1srxe eraql ueqtr,

r=! r=f.,C-S=oS,C-A =oA,CAn!,V))=,C'(gnlil)= "

66

1as '.raq1rng '

,3 ro1

lurod aseq aqt eq op wj

'salrnc qloous pasolc aldurs "t" t=;{ llal'l!,vjI = ,K

pu" I=r{ llAl'llV)l = Z ur fg pue '!,V'lg 'ff

I1e teqt aurnsse fetu am'1srrg'J ue qcns go

uorlcnr?suor e earE 1leqs ea\ 1nq 'lceJ u^\oul-ila/( € sr sIqJ 'l(6-)-l',gj = [,9',9]

qcrq^\ roJ S: oluo A p { ustqdroruoeuroq Surnresard-uorlelueuo u" slsrxe ereqluJ > lB

'g] fue roJ teql /$oqs ol lu?Icgns sr 1r 'f1r,rr1ca[rns eql a,rord o;,

'16' ,Sl = [/

',S] eleq e.rlr snqa 'ctdolouoq er€ rd Pue / feqt flrsea a,rord

uec a \ 'aue1d aql ul {s!p lrun aq} o1 crqdrouroeuoq $ U urorJ fg pue fy 1e Eur-1e1ap fq peurc?qo uretuop aql aculs 'uorlrusap aqt fq [td'S] = [6',9] teqt atoN

9I6 snuag;o aeedg rallgurqrral 't'I

16 1. Teichmriller SPace of Genus 9

we find a qua.siconformal mapping /o homotopic to / (Bers [26] or Lehto [A-68], Chap.5, Theorem 1.5). This fo is not necessarily smooth; however, there

exists a real-analytic quasiconformal mapping homotopic to f, (the Corollary to

Theorem 6.9).

Finally, we define a canonical group action on the Teichmiiller space ?(R).

Let Mod(R) be the set of all homotopy classes [o] of orientation-preserving

diffeomorphisms ar: .R * -R. We call Moil(R) the Teichmiiller modular group or

the mapping class group of .R. Every element [ar] acts on ?(R) by

[r].([S, /]) = [S, f or-']

for any [S, /] e "(n). We call every lw)* a Teichmil,ller moilulor transformation.

Let Mo be the moduli space of closed Riemann sarfaces of genus g, i.e., the

set of all biholomorphic equivalence classes [S] of closed Riemann surfaces ,9

of genus g. since for a,n arbitrary closed Riemann surface .s of genus g there

exists an orientation-preserving diffeomorphism of R onto ,S, the moduli space

M, is identified with the quotient space T(.R)/Mod(R) of "(i?) by the action

of. Mod,(R). Therefore, we can study the moduli space Mo via the Teichmiiller

space ?(.R) and the Teichmiiller modular group Mod(R). In Chapter 6, we shall

see that "(E) has a (3c - 3)-dimensional complex manifold structure and that

M od(R) acts properly discontinuously on "(8) as a group of biholomorphic

automorphisms. In particular, the moduli spare Mo has a (3g - 3)-dimensional

normal complex analytic space structure.

1.4. Quasiconformal Mappings and Teichmiiller Space

Let us review the Teichmiiller space ?(,R) constructed in the previous sectionfrom the view-point of the theory of quasiconformal mappings.

L.4.1. Deformation of complex structures and Beltrami coefficients

For a point [S, /] g ?(,R), we want to compare the complex structures of ft and

s. Take a coordinate neighborhood (u,z) on I and a coordinate neighborhood

(lz,to) on ^5 with f (U) C V, and set F = ?r,ofoz-l. Then

p -

is a smooth complex-valued function defind on iur open set z(Lr) in the complex

plane. Note that it is independent of the choice of a local coordinate u.'. Since

] i, .n orientation-pr"r"ruing diffeomorphism, the Jacobian of F, i'e., lF,l' -

l&12 i" positive-definite on z(U). Thus we have lpl < 1 on z(U). Further, F' is

biholomorphic on z(U) if and only if F = 0 on z(U)' We call y' the Belt'rcrni

coefficient of / with respect to (U , z).It should be noted that a Beltrami coefficient of / depends on the choice of

a local coordinate z on R. How it depends is shown as follows: take coordina*e

Fz

F'

.l(o)'rl-I -i(g;ffi=(o)>r

u asdrlla srr{} Jo srxe rounu aqt o} srxe roleur aq] Jo ol]er el{l

'l"l (l(o)'rl + r) l(o)"/l i l(o)zl 5 l,l (l(o)'tl - r) l(o)"/lsarlrlenbeut aqt ,tg '(tt't '8t"f)

aueld-rn aqt ul asdrlla ue o1 aueld-z aql ur 0 raluec qt-ra elcrlc e spues 7 deurr€aurt eqt 'raaoero141 't > l$)"t /(O)ttl = l(g)r/l pue O * @)"1 1eq1 saqdun qcrq,ra

'o < .l(o)"/l - .l(o)"/l = (o)/r

sagsrles 0 - z le (6)f uerqocel s1t 'urstqdrouroagrp Sutrlraserd

-uorleluerro ue sr / acurs '0 - z Ie / ;o uorsuedxa ro1,te; eql Jo tural rapro lsrgeqt aq z(iltt + z(g)'t = G)l 1a1

'aue1d-rn xalduroc eqt ul /O uIPruoP e oluoaueld-z xaldruoc aql ul 6 urSr.ro eq1 Surureluor O ureluop € Jo tuuqd.rouroagrpSurarasard-uorl€luarro ue sr / l€ql etunsse arra 'spooqroqq3reu aleurpJoo? Surraprs-uoc 'srql eas oI 'sluer)lgeoc tusrtleg;o Surueeur cr.rlatuoa3 eq1 ureldxa a.tr '1srrg

s8urddetr4l lBruroJrrocrsen$'6'7'1

',t1t1eur.royuoc uorJ 3[ 3o uotletaap aqlernseaur o1 pesn sl / Jo luerrlgaof, lruprllag aql pue 'g uo arnlrnrls xaldtuoc eql

Jo uorleruroJep e sluasardar (y)"6 ul U'S] lurod e leql su€errr lr ef,uag '(U)"f ul

Wl'lAl = [/'^g] ?eql s^roqs srq;'Surddeur crqdrotuoloqlq € q,S * {y :l 7eq1pue 'tusrqd.rouoeJrp Surl.rasard-uolleluelro ue sr /U * A :p? deu flrluepr eq1

1eq1 's1as se U = /U leql eloN 'tl)D{ (Io"*'("1)r-t) } spooqroqq3rau eleurp

-rooc ;o uals,ts qlurr paddrnba IU ateJJns uu€uIeIU A\eu e aleq ea,t 'deilr slqt uI'U uo arnlf,nrls xaldruoc " seugep v>a{(toDm'("1)vt) } spooqroqq3rau eleu-rprooc ;o ualsIs B ',9 - g : rt ursrqdrouoeslp Sut,uasard-uolleluelro ue roJ

puts S uo vl"{ ("*'"A) } spooqroqq3rau eleutp.rooc Jo ualsds e rog '.Lrop

(e'r )

'U uo / 1o Tuata$aoc naDr?Ieg eql pallec $ q)nl^t

''P ,t - trlzp

,tq ,{ldurrs (t't-) ed{1 ;o ruroJ l"rluereJlp slq} a}ouap e^r snq;,'U uo (I'1-) ad{1 Jo urroJ leltuareJlp € se)npul U Jo spool{ro,Q{31au el€ulproocuo /go sluerrlgeoc rur€rllagJo les eqt leql s.lroqs slql'rizors - l{z areq^l

(r r)'(trut2)tz uo (#) l@).(rzotrl)= trl

e^tsq a,lr 'Q * qnU ln

uaq,11 ',,(learlaadsar '(qz'qn) pue (lz'fn) ol leadsar qll/'^ ./ Jo sluarf,lgeoc rurerllageqt eq 'trl pup ld lr,1't1 1 (r2)l pue ln > (dI ?€qt qcns g p (tn'tn1'(!m'11) spooq.roqq3reu eleurprooc pue gr 1o (tz'qn) '(tr '.!2) spooqroqq3rau

L1a*dg ralnurqf,ral pu? s8urddul4l purro;uorrsen$ '7'1

18 1. Teichmriller Space of Genus g

This shows that any infinitesimally small circle with center 0 is mapped by / toan ellipse whose ratio of the major axis to the minor axis is K(0).

L ( z )--t

o: la reu$) cp - 0+aref,Q)a : (1+ lp (o) l ) r l l , (o ) lb : ( r - l p ( 0 ) l ) r l f , ( o ) l

Fig. 1.11.

This statement holds at every point in D. Thus we also call the Beltramicoefficient

, \ f t ( r )p tQ)= f f i , ze D ,

the complen dilatationof / at z. As we saw before, Ft = 0 on D if and only if /is a biholomorphic mapping on D. We call f a quasiconfonnal mapping of D to

Dt if f satisfies

Kr = sup l* lrrl 'J! . .".' , e b r - l p t ? ) l

Further, f is called a quasiconformal mapping with Beltrami coefficient Lrt.W"call K1 the maximal d,ilatation of f .

In this chapter, we only consider smooth quasiconformal mappings. We shallstudy more general quasiconformal mappings in Chapter 4.

tansformation formula (1.4) implies that the absolute value lprl(z)l of the

Beltrami coefficient W = pJQ)dzldz of an orientation-preserving diffeomor-phism I : R - .9 does not depend on local coordinates on l?. Thus lpy I is acontinuous function and lpty | < 1 on ,t. Since r? is compact, we get

ln particular, we have

l lpr l l - = sup lpy(z) l < 1.z e R

(1 .6 )

"'=::B HP)t=il+ffll:'* ( 1 .7 )

'@)"ttpO o7 s|uo1aq nP Quo puv l? A lo dout fi1t7uapt aq7 o7 ctiloTouroq sr,

lorlor-6'atou.t.r,eql,.rng '(A)+ttlO tt, oeluos.r.ot sp1ot1 (d)*o- 6rl uotTo1et' aq7

{r fi1uo puo t? rS - S:r1 Autddout ctyiltou,rolotpq o sqrne uall'tS + f, i0

puo S * A i{ stusttliltoutoa$tp 0untasa.td-uotToTueuo rof, '9'I ureroaqJ

(o'r)

'r(A)g Jo luauele ue sr /r/ pus (U)+//?O Jo luaruale ue sI r't areq^l

, / Iil:a-I g) = r-.o[rt = ({rt)*n 'vl_no\-ril:lrl r.)-

""-\

fq ua,u3 q r(2,)g uo (A)+ttlO Jo uollc€ aqJ.@)+l

lgO;o dnor3qns

Ieurou e q (A)'! !16r '{1.rea13 'pg deru flrluapr eq} ol ctdolouroq (u)+l tlo

ur s?ueutrrela II€ Jo slflsuoc qclqar dnor3 e aq, (U)"t lpO leT JIeslI oluo Ur uo

sursrqdrouoagrp Suu.raserd-uorlelualro IIe Jo dnor8 aq1 (A)+t tlO fq elouag'(g't) fq uarrrS urrou-oo? eQt Eursn {q t(g.)g uo f3o1odo1 e eugeq'seceJ

-rns uueureru pesol, oluo ar ec"Jrns uuerualu pesolc pexg e;o sursrqd.rouoeJlP

Surlrasard-uorlplua-Iro II" Jo stualrgao? Itu€rlleg Jo les aqt aq r(U)g larl 'U uo

sluer?lgeoc lru€rlleg go aceds eq1 Eursn dq saceds rellntuqclal replsuof,eJ sn lerl

sluarcsaoC rtuertlag 3o sacedS '8't'I

'seceds rellnuqtlal go froaql t.Ildleue xelduroc e dole,rap eirr eraqm '9

reldeq3 ur elor luelrodur ue sr(e1d slqJ'tr/ uo rtllertqdrouoloq spuedap /orrl

luarclsao) rurcrllag aqt'./ paxg e roJ l€q? suoqs (g'1) elnruroJ'ta,roaro141'((g'Z) "lnurroJ'9'6 eurural) y ut lutod

e fl o pu" requnu Ieer " $ d aleq^r 'aue1d xaldruoc eql ul y cslp ?tun eql Jo

ZD_T

- "a = (z)L

ursqdrouolne ctqdrouroloqlq e se ruroJ eruss aql seq (8'I) "1nlurod 'tlrDu.Iay

.z{il --'trt fi npo puo lt ctyifuoraopqNq sp zg + tg :r}lozl |urildou.t eW'(7,'l

= !) lS 1- U : lt sutstrld.tou.toa$tp |utatasatd-uotToTuauo .tot '.t'o7nct'7-tod u1'spptt

(a'r ){ooyl.[il-I'l _ r^oe -H:j;Frt

T-t-"

uotlllar eqt 'J + g :6 'S ,- A : t su,tst'r1d.t'ou-toa[9p|um.tasand-uorlDlueuo puD 'J'g 'g secol.tns uuour?tg Jotr 'g'T uollrsodor6

'elnl uleql eql ,tq pe,rord ,,(lsee

sr Surddeur elrsodtuoc e Jo sluelcseoc rtu€rlleg roJ elnuroJ 3ur,u,o11o; eq5'g

;o Surdderu l€ruJoJuoctsenbe s-I ,g - g, : / usrqdrouoe$IP Sur,r.reserd-uoll€lueuo ue leql {es feru ai ecuaH

6Iacedg relnurqf,reJ Pu? s8urddeyl TeurroJuof,rsenb 'l'f

20 1. Teichmiiller Space of Genus g

Prool. Suppose that there exists a biholomorphic mapping h: S * S'. Settingu - g-Lohof, we see that formula (1.8) gives

Fg = FhoJo r - r = PJou - t = u * (P t ) .

Conversely, if there exists an element u e Dif fa(R) with pc = u*(pt),thenProposition 1.5 shows that h - gowof-r: ,S * S'is a biholomorphic mapping.

The second assertion is clear from the definition. D

Corollary. The mapping of sending (S,f) to pt e B(R\ iniluces lhe followingidentif,cations:

r(R) = B(R\lDif I"(R),Mo e B(R)r/Dif f+(R).

1.5. Complex Structures and Conformal Structures

In this section, let us reconstruct the Teichmiiller space ?(R) by means of con-formal structures induced by Riemannian metrics on R.

1.5.1. Riemannian Metrics and Conformal Structures

Suppose that a Riemannia.n metric ds2 is given on a real two.dimensional ori-ented smooth manifold M. This metric is represented as

ds2 = Ednz -f2Fdxdy * Gdy2

on a coordinate neighborhood ([/, (r,V)) of M. SettinE z = t f iy, we see thatit is written in the form

ds2 = \ ldz * prdZl2, ( 1 . 1 0 )

where ,\ is a positive smooth function on U and pl is a complex-valued smoothfunction with lpl ( 1 on U. Actually, l and p are given by

^= i ( " +G+zJ -nc - r ) ,E _ G + 2 i F

t ' - E+G+2\/EE=7t '

Local coordinates (u, u) on Lr are said to be isothermal coordinate.s for ds2 if

ds2 is represented asd s 2 = p ( d u z + d u 2 ) ( 1 . 1 1 )

on U, where p is a positive smooth function on [/. Here we €Issume that both ofthe orientations induced by the coordinates (c, y) and (u, u) on U coincide withthe one on M. The complex coordinate u) = 1r * iu is also called an isothermalcoordinate for ds2.

fq uarr3 q ;4I uo (z'fr'r) - d lurode'T,I'I'3ld rl pel"rlsnllr n (d'il seleulproot I€col aql 3urs1 'r11 uo )ulauru"ep{?ng eqt dq pef,npur zsp crrlau ueruuetuelg Ietluouec € s€q J,tI ueqJ

'(Zt't'ft.f) D > q > 0 areq^4,'srx€ z eql punor€ aueld-(z'f) "qt uo zg = zz * z(D

- f) ala.rrc aql 3u-I^lo erfq paurelqo fl r{?rqr'r gll af,Bds ueaprTcng aql u! e?eJrns e aq W p1 'aldutoxg

'droeql uollcunJ ctrlatuoa3 aql papunoJ pue drqsuorleler slql paztuEo-rer lsrs uu€uerg 'g 'ses?c

leuorsueurp .raqSrq roJ enrl lou $ qtlq^\ 'sployrueur

Iear l€uorsueurp-A ''e'l 'sp1o;rueur xalduroc I€uorsuaurp-1 .ro; ,tl,radord elqe4reur

-eJ e sr uorlrasse stq; 'Surdderu l"ruJoJuor € pells? sr Sutddeur cqdrouoloqtq

e teql uos€er " $ srqJ '1ua1e,rrnba are aln?cnrls IsruroJuo? Jo pu" aJnltnrlsxaldtuoc ;o sldacuoc 'esea

leuorsuaurp-oirrl aql q (teql s^\oqs uaroeql slqJ

'att1d.rouro1ot1lq s? S *A tl t! fi1ao puo lr lout^,t,otuoc s! ([sp '.,1I) - Qsp'W):l

uaql 'fi1aar7cedsar '([sp'N) puo (zsp'W) splottuotu uvruuDutary louorsueurrp-Z pa?ueuo frq pacnput, sacottns uuou?tg eq S puv A pI 'Z'I uraloaql

'ureroeql 3ur,no11o; eql ol speal (61'1) uoll"lues-e.rda.r aq1 go ssauenbrun eql 1eql aas o1 fsee sl ll 'U uorsuaurlp Jo esef, ar{l uI

'uaql uaa&rlaq Surddeur FluroJuols slsrxa ereq? J! ernptury loru.totuoc euros eq1 eler1 ro Tuapatnba Qlou.t.r,oluocare ([sp'N) pue (zsp'W) 1eq1 {es a1l'N uo (zC)t pue (t5l;;'ueaallaq'lspfq pa.rnseaur 'a1Eue aql slenba W uo 7'C pue I, selrn? qloours fue uaeiu,leq'zsp fq, pernsperu 'e13ue aq1 leql srrcetu 1r

'f1arrr1rn1ul 'W uo uorlcunJ rllooruspenle^-lear v 4 6 araq,n '_;211 uo .sp(d)dxa o1 lenbe q ./ ,(q ltp to lceq 1ndeq1 yfuttldnutputlotuoc"slN*WitusrqdrouroegrpBur,rrasard-uorleluarroue '(|sp'N) poe (csp'A[) sploJrueur ueruuetuel]r leuolsuetulp-U palualro rod

'zsp clrleru u€ruueruerg eql fq pecnpw ?rnpnr?s lout^to{uoc eql palle)aq ,(eu U uo arnlcnrls xaldruoo eq; 'deu, slql q pautelqo aceJrns uueurelu eqlg itq eloueq 'W uo ernlcnrls xalduroc s saugep r>!{(!n'h)} l€q} f;r.re,r o1qncJlp lou sr lI '!2 qcee uo lm elsurpJoo? I€r.uraqlofl ue slsrxe araql (,;ig;o

r)!{((n'!x)'12) } spooq.roqqErau eleurprooc Jo uralsds e ro; 'acua11 'I > -llr/ll

?eql paphord slsrxa s.ile,rale 01 uorlnlos " qcns '7 .ra1deq3 Jo A$ ul pe,rord st sy'uo4onba ,urDrlleg aql pell€r sr uorlenba slql 'rn uorlnlos crqdrotuoaglp e seq

(elr)9a=9m8 m8

uorlenba prluereJrp lerlred eql JI s?s1xe

esp roJ /'1 al€urprooc l€uraqlo$ ue leql apnlcuoc erra'(0I'I) qlrrn Sur.reduroc

tzotrl ' ,lzn i+ zPlzlzmld = "lnnld

sagsr??s esp roJ nl eleurProoc leruraqlosl u3 a?uIS

sernlrnrls I"ruroluoC pu€ sarnlf,ulg xalduro3 'g'1 tz

22 l. Teichmriller Space of Genus g

F ig .1 .12 .

, = (o * 6cos g )cos0 ,

y = (a * Dcos<p )s ind ,

z = bs ing.

We assume that the orientation of M is induced from (d,g). If we set

,1, = rl,@) = [' ---!- 4o,J o A + O C O S / P

then the metric d,sz = (a * bcos g)2de2 + b2dgz has the form

d s z = ^ ( { ) ( d 0 2 + d r l } ) .

Thus to = 0 + fry' is an isothermal coordinate for ds2 on M, which defines acomplex structure on M. Hence ft is a torus. A little more calculation showsthat R is biholomorphic to C/f, where l- is a lattice group generated by 1 andib/\/i, -F.

L.5.2. Reconstruction of Teichmiiller Spaces by Riemannian Metrics

Fix a closed Riemann surface r? of genus g ) L. Take any local coordin ate z onR.

For an arbitrary Riemannian metric ds2 on rR, from the uniqueness of theexpression in (1.10), we obtain a globally defined Beltrami coefficient p on R,being a differential form of type (-1,1) and llpll- < 1. Such a p is called theBellrarni coefficient induced by a Riemannian melric.

Let us observe the relationship between the Beltrami coefficient of anorientation-preserving diffeomorphism and the one induced by a Riemannianmetric.

\ a

'@)+l lvo /(a)w = uw'@)'l lto l(a)w = @)t

:suotToc{zyuapt, 6urmo11ol ayy aat f

{ o7 0utpuodser,ror )uleru o to'fr1aa4cadsa.r,'sso1c acuapatnba 6uo.t7s puD nuelD-amba aq7 ot (A)Ju! lt'Sllueuep uo pues qnqm sfurddvu eqJ 'g'I uraroaqJ

'uorlress€

3ur,rao11o; eql of p€al g'I ruaroeql o1 ,t.re1oro3 eql pu€ uorle^Jasqo srql',,t1e,rr1cedser'(A)W p sass'elc acuaprr,rnba 3uor1s IIe pue ecuele.,lrnbe IIeJo las eq?

@)"ttpo/(A)W pue (a)+IlpO/(A)W rq eloue6l '(A)"lIlO o1s3uo1aq o srqly Tualoamba fi16uo.t7s eq o1 paugep a.re |sp pue u sp 'reqlrnJ 'l€ruroJuoc q (I*p'U)- (zsp'A),o ?tsqt qrns (A)+ttlO ur r,l lueruele ue slsrxe areqlJr Tualoaznba aq01 paugap erc (g)4r ur |sp pue 6s'p slueuale o rI 'g

Jo (U)J eceds rallnurqcraaaql llnrlsuocer a^r '9, uo scrrlaur u€ruueurelg Jo (U)hf les eql Sursn 'no51

'w uo s)rrleru ueruu€ruaru ,{q parnputsluarrlg:aof, ttu€rllag Jo las aql ol lenbe sr Ur uo srusrqdrouroagrp Surarasard-uorleluarroJo sluer)Ueo, rurerllagJo I(U)5. ?es eql leql ees a,r.'.{e.n sqt uI

't o7 0utpuodsatloc A uo culeur uvruuoulerye eq ol pr€s sr (|rp)-t crrlaru e qcns'S'Jo ernlcn.rls xalduroc eq1 Surcnpur [spcrrleru € Jo ecroq) eql uo puadep lou saop uotlress€ slql l€r{l eloN '/

Jo }€r{t se

luel)cgeor rur€rlleg erues eql se,rr3 / .repun |sp l" (trtp) -l >1ceq 11nd eq1 uaql '(g

raldeq3 Jo t'I$ 'gc) ,t1a,rr1cedse.r 'crrleru er€rurod .ro 'ueaprlcng 'pcrraqds eql ,(q

pe)npul sr qcrq/( S' uo crrleu eql e{€t alA '(U .ra1deq3 Jo 6'Z ueroeqa) eueld-;pqraddn aq1 ro 'aue1d xalduoc eql 'araqds uu€ruerg eqt ol crqdroruoloqlq sl S Joac"Jrns Surra.roc lesrelrun aq1 '1ce; uI 'SJo auo leur8rro aql ol lual€^rnbe sr Ltp fqpa?npur arnl?nrls xelduroo eql leql os S' uo |sp or.rlaur ueruu€ruerll " a{€l uer eM'ue,rr3 eq .g * U :/ ursrqdrouoegrp Surrr.raserd-uorleluerro ue 1e1 'd1asra,ruo3

'rl qtl* saprcuroc luercgeoc rureJllag sllpue ursrqdrouroagrp Surrr,.rasard-uorleluarJo u€ sl ,U * g. : { Eurddeur ,t1r1uepraq? ueqJ '(^'n) eleurproof, leurreqlosl aql ^q pernpur sr arnlcnrls xaldtuocasoq^r /U a)eJrns uuer.uerg e urclqo all.'(z 'p) pooqroqq3rau eleurpJoof, q)Ba uo

uorlenbe rurcrlleg aql Surr'1os 'I'g$ ul uaes ueeq s€q sV 'A to zsp rrrleurueruueruerll e ,tq pecnpur luerrlgeor rtuerllag eql s! r/ 1eq1 esoddns '1srrg

zo zo --:- fl = ::-m8 n8

8,2sarnlf,nrls l"urroJuoC pu" sarnl)nrlg xaldurop 'g'1

24 1. Teichmriller Space of Genus g

Notes

The geometric function theory originated with Riemann's 1851 Gottingen dis-sertation [181] and his 1857 paper [182]. In connection with multi-valuedanalytic functions such as algebraic functions, he introduced the concept of theRiemann surface as a branched covering surface over the Riemann sphere. Healso recognized clearly the intimate relationship between holomorphic functionsand conformal mappings on a domain in the complex plane. In [181], he proved

Riemann's mapping theorem which asserts that any simply connected domainin the complex pla.ne with mor'e than one boundary point is biholomorphic tothe unit disk. In [182], he obtained the Riemann-Roch theorem. By using thistheorem, he determined the degree of freedom of finite branched coverings overthe Riemann sphere which represent closed Riemann surfaces of genus g, andhe obtained the complex dimension mo of the moduli space of closed Riemannsurfaces of genus g, that is, ms = 0, 1, and 39 - 3 for g = 0, l, and g > 2,respectively. For more complete exposition of Riemann's work, we refer to Ahlfors

[a] and Klein [A-53].The standard definition of a Riemann surface, that is, a one-dimensional

complex manifold was introduced for the first time in Weyl's classic [A-111]"Die Idee der Riemannschen Fl6che" in 1913.

The material of this chapter is classical. Some of the many celebrated bookson Riema.nn surfaces are Ahlfors and Sario [A-6], Bers [A-13], Cohn [A-22],Farkas and Kra [A-28], Forster [A-32], Griffths and Harris [A-39], Gunning

[A-40], Jones and Singerman [A-48], Schlichenmaier [A-95], Siegel [A-98], and

Springer [A-99]. For details of topology on surfaces, there are further books byBirman [A-18], Harvey [A-4lj, Chapters 1 and 6, Moise [A-75], Stillwell [A-101],and Ziescha,ng, Vogt and Coldewey [A-114]. For algebraic curves, we refer tothe books by Arbarello, Cornalba, Griffiths and Harris [A-9], Grffiths [A-38],Mumford [A-78], Namba [A-82], and Shafarevich [A-97]. The moduli space oftori considered in $2 isstudied in the context of elliptic curves, elliptic integrals,and theta functions. For an interesting exposition on this subject, see Clemens

[A-21], and Jones and Singerman [A-48].For textbooks on Teichmiiller spaces, there are Abikoff [A-1], Ahlfors [A-2],

Gardiner [A-34], Harvey [A-41], Krushkal' [A-60], Lehto [A-68], and Nag [A-80].For expository papers on this subject, consult articles by Ahlfors [8] and [11],and Bers [22], [29], and [40]. The approaches to Teichmller spaces as in $4 and $5are found in Earle and Eells [62], [63], and Fischer and tomba [7L], respectively.

'H Jo 'J '3 secottns uuDurelya?rtn e1t lo auo o7 Tualoatnba Qparydloruoptfq st acoltns-uuouaty peloauuo?fi1du.tzs fueag (aqeox pue ?recurod

'.r!"IX) 'tuaroaqtr, uoltBznuJoJrun

'(w6d '[y-y] sroylqy eas) {$p }run aq} ot crqdrouroloqrqsr lurod f.repunoq euo u"q? arotu qlra C ur ul€urop pelceuuot fldurts dra,ra

1€rll slresse qcrq/rr uraJoall ,utdrlou,t s.uuDur?ta ol Pa)nPer sI rueroaq? sHl'caueld xelduroc eqt ur sur€urop ro; 'r(gercadsg 'sac€Jrns uu€tuerlf roJ sploq uaro-eql uollezlruloJlun eql Pallec sl q?lq,$ ?oeJ alqe{r€IueJ e }sq} u^lou{ 11am s! 1I

rrreroaql uorlBzlruroJlun'I'z

'6 snuaS;o tg acedsrellnurq)Iel eql ql!,u PagluePl q ll }"qt alord pue'g-ngll uI lasqns e se td

aceds alcr.rg aql eu$ep aal '(6 l)f snueS 3o sereJJns uu€r.uarlr pasolc Surluasardar

sdno.rE uersrl?ndJo s.ro1e.raua3;o suelsfs Ierruorr€c Eursn'g uorltas ur 'fgeurg'ralel pasn ele qtlq^\

'sdno.r3 uersqcr\{ yo satl.redord freluaurala euros a,rord airrr '7 pue t suolltes uI'; dno.r3 u€rsrlf,nd e Aq H Jo J/H aceds luatlonb e fq paluasardar

q (Z ?)n snua3 ;o ereJrns uuetuerg pasolc fre^e leql epnl?uot a,rl 'dem srql uI'dnor3 uersqrnd e J II€r ell'' 11 = U uerl^r 'relnarlred uI 'suolleuroJsueJl snlqgl tr;o Surlsrsuoc dnor3 e sB g uo rtlsrionurluo?srp fpadord slce J pu" 'Il ro'C '?

o1 lualelrnba fllecrqd.rouioloqlq sr g, 'ruaroaql uorleznuJoJrun eql fg '6 uorlaaiur palrnrlsuoc s-r J dnor3 uorleur.r5gsue.rl Surra,roc sll 'U

Jo U eceJrns Eutreloc

Iesra^run eql JePrsuo? arn tg, aae;rns uueuaru frelrqre ue ol ueroeql uollezfluroJ-run eql {1dde o1 repro uI 'g aueld-g1eq .reddn eq} ro '9 aueld xalduroc eql 'e

a.raqds uu€rueq aql :sae"Jrns uueruenr eerql eql Jo euo o1 crqdrouroloqlq $ eceJ-rns uusruarg palreuuoc fldurrs .,(Je a leqt slrass'e qcrqilr 'aqaox pue 'atecuto4'ura1y o1 enp rueroeql uorlsznuroJlun aql urc1dxa er,r '1 uotlcag u1 'sdno.r3

u€rsqrnJ pue 'suorleurroJsueJl snlqgl [ 'sece3:tns Surralroc lesrellun uo s]c€J f,Issq

'sace3:rns uu€ruerg Jo ruaroaql uort"z-turoJlun eql apuro.rd a,u 'asodtnd sn{l roJ'd snua8

;o eceds e{llq aq} pag"c fl rlcrq^{ (e-rsg ur }asqns e se pezrper sr (6 l)f snua3

;o aceds rellnurq)ral eql l"qt ^{oqs ol u .raldeqc luesard eql Jo esodrnd aqa

ar€ds a{rlqjt

u raldBrlc

26 2. Fricke Space

Remark. These Riemann surfaces 0, C, and /y' are not mutually biholomorphi-cally equivalent. The Mobius transformation tr = (z -i)lQ * f) maps biholo-morphically 11 onto the unit disk 4, and hence we often use the unit disk 4instead of the upper half-plane If .

Corollary. A closed Riemann surface of genus 0 is biholomorphic equiualent tothe Riemann sphere e . Thus the moduli space Ms of closed Riemann surfacesof genus 0 consists of one poinl.

Proof. Since a closed Riemanil surface R of genus 0 is simply connected, theuniformization theorem implies that .R is biholomorphic to one of the threeRiemann surfaces e^, C, and 11. Since .R is compact, it should be biholomorphi-cally equivalent to C. !

For proofs of the uniformization theorem, we refer to books on Riemannsurfaces listed in the notes of Chapter 1. See also Ahlfors [A-3]. For historicaland expository accounts, see Abikoff [2], and Bers [29] and [36].

Among standard proofs for the uniformization theorem, there is a method

in which a mapping function is constructed by using Green functions. Let us

elucidate the concept of Green functions by using an intuitive example fromelectromagnetism. We rega.rd a Green function on a Riemann surface .R as theelectric potential on R where a positive charge is given at a point p and whoseboundary is earthed. Mathematically, when z is a local coordinate around p

on R, we define the Green function g(.,p) on ,R with pole at p as the minimal

function in the family of positive superharmonic functions which are harmonic

in .R - {p} and have the singularity -loglz - z(p)l at p. The existence of aGreen function depends on "capacity" of the boundary of rt and is indepent ofthe choice of a point p. For example, the Green function on the unit disk 4 withpole at zo is given by logl(l-z;z)/(z - z")1. On the other hand, there are no

Green functions on C or C.Now, assume that there exists a Green function g = g(,p) on R. Then we

obtain a biholomorphic mapping f : R'-'+ A;

f k) = exp (-s(q) + ic.k)), ( 2 . 1 )

where g* is the conjugate harmonic function of g on R - {p }. Note that g* is amulti-valued function whose periods are 2ntr (n e Z) because of the singularity ofg and simple connectedness of R. Hence, / itself is a single-valued holomorphicfunction on .R. By the argument principle we see that / is univalent, i.e., a

biholomorphic mapping of -R onto 4 (see Fig. 2.1).Next, we deal with the case where there exist no Green functions on R. We

take a sequence { n' }Lr of simply connected subdomains of E such that .R' is a

relatively compact subset of E +r for each n, that Uf=rR" covers .R except for at

most one point, and that every ftr has the Green function g,. with the commonpole p. Then, in the same way as before, we can construct a biholomorphic

mapping fn: Rn - Alor every n. By multiplying each /' by a suitable constant,we get a sequence of biholomorphic mappings F' : R- - { tr e C I lr.rll < r' } so

I€sJelrun y 'U Jo acottns |ut.taaoc lDsrearun " U pue'g 1o 6uuaao) IDsrearun e

(A'o'A) II€) e^\ 'palcauuoe flduns sr U uaqm '.raillrnd 'U otuo g 1o uo4cato"td eq|pa11ec 6qe sr z deur Surra,roc aql 'Ujo acn!.rns |uuaaoc e g iue

'g 1o |uttaaoc

e (A'v'U) IIec aaa 'r1ooq srql uI 'crqdlouroloqlq q /? * ]1 :t deur palcr.rlsaraql '11 p (A)r.-" a3eun asra.rlur a{l;o 1 luauodruoc pe}cauuor qcee roJ leqlqcns , pooq.roqqSreu e s€q A Jo d lurod fra,re y dout, Fuueaoc e eq ol pres sr

A - U : ! Surddeur crqdroruo-Joq anrlcalrns y 'sac€JJns uueuaru aq Ur pue Ar ]erl

sdno.rgrrorlerrrroJsue4l, Eurre^oC puB saceJrns Eurraaog Jo suor+Iugo1, 'TZ'Z

'e?sJrns uueruarg f.rerlrqre ue Jo aceJrns Surrarrocpelceuuoc fldurs € lcnrlsuoc aaa 'uraroeql uorlszrruJoJrun eql ,f1dde o1 rapro uJ

stur.ra,ro3 lesJa,rrun'e'Z

'c ro c oluo u;o Eurddeu erqdrotuoloqrq parrsap e sef,npur 'Ur*n uo uorl?unJ fr-uliaqf leql

'r'z'8tJ

fldurrs e

[€ruJoJuot--

lrnf /1 So1 - (*)"6;o qder3 ((4t)"0 - (r)6 uort?unJ uaarg erlt Jo qder3

lr>lmll: v

II+--- I

LZsBurra,rog lesrel-ru1'Z'Z

28 2. Fricke SPace

covering of R mea,ns that it is the "highest" covering surface of all coverings ofr? (cf. Theorems2.2 and,2.4, and the remark in $2.2).

Example I. We give a few simple examples of covering surfaces.

(i) Let r: C - C - {0} be given by r(z) = e'. Then C is a universal covering

s u r f a c e o f C - { 0 } .( i i ) Let r : H + A-{0} be g iven by r (z) - e2 ' i " .Then I / is auniversal

covering surface of A - { 0 }.( i i i ) Let r : C - {0} * C - {0} be g iven by r (z)= zn, where n is a posi t ive

integer. Then C - { 0 } is a covering surface of itself, but it is not a universal

covering surface.( iv) For a g iven ̂ (> 1) , set r = exp(-2r2/ log, \ ) and A= {w € C l t < l . l <

1) . Def ine r ; H-- -+ Aby r (z) - exp(2tr i logz/ log. \ ) , where logz denotesits principal branch. Then 11 becomes a universal covering surface of the

annulus ,4.( v ) t e t 4 bea la t t i ceg roupgene ra ted by 1 andapo in t r € I / , and le t r be

the projection of C onto the quotient space C f fr. Then C is a universal

covering surface of the torus C/ lr.

Any biholomorphic mapping '1 , fr, - E wittr ro^l = z is-called a coaeringtransformalion of a covering (R,r,R). For a given covering (R,r, R), denote by

l' the set of all its covering transformations. By the composition of mappings,

l- forms a group, which is called the coaering transfonnation group of (,R. r,,R).

In particular, we call I the uniaersal coaering lransformation group of (-R, r,.R)

if ,R is a universal covering surface of r?.

(i)'(i i) '

(iii)'(iu)'(") '

f = (rr) with 71(z) - z *2tr i .f = ( r r ) w i t h 7 1 ( z ) = z * 1 .1= ('rt) with 71(z) - z exp(2riln),which is a finite group of order n.

r = (zr) with 71(z) = )2.7 = (T,72) -- f,, where 7{z) = z * 1 a,nd "fz(z) = z + r.

2.2.2. Construction of tJniversal Covering Surfaces

First of all, we need several definitions. A path on a Riemann surface R means a

continuous curve C: I - R, where.I is the interval [0, 1]. The points C(0) and

C(1) are said to be the initial and lern'inal points of C, respectively. We also

say that c is a path from c(0) to c(1). Throughout the book, if no confusion

is possible, its image C(/) is also denoted by the same letter C.

fdolouroq € eleq am'f11eutg 'I ) n roy (n1)"g = (n)(c't)p fq g'uo (t't)p qlede ausep a,r'7 x / f (s'?) fue ro3'uaqJ,'1 3 s due .lo;'d = (I)"f, = (0)'Jpue 'C = rI 'oI - og sagsrles tl'{ ("'.)uf = 'd

} leql q?ns uaql uee^.rleqrldolouroq € eq U * 1 x I ii p1 'o7' o1 crdolouoq sI C leql su€etu q?Iqtl'fod'Cl --

lod'ollleqt epnpuoa am 'q1ed pasoll " $ C ecurs 'l"d'Cl,tq peluasarde.r

sl C Jo lurod leurturel eql puie 'od lutod es€q IIII^a g uo qled Pasole e sI 5t uaql

'ei" - C tnd

'[od'olTol crdolouroq q [od''I] lutod eseq q]p\ g uo g qled Pesopfierta 1eq1 aes ol luelrgns fl 1I

'Paleeuuos ,(ldturs q !f f€qt b,rord arr,r ',r,r,o11

'pelceuuoc q U leql satldtut qclqar '[d'g] ol lod'oll uro+ U uo I qled

€ a^eq al,r '7 ) s fle,ra iog [(s)g'"C] = (s)g 3ur11ag 'I ) 1ro; (ls)j = (rt"C rq

A uo'C Wed e eugep'1 I s q?ea ro,{'U uo qled e Aqlod'o1f ql.I^{ pal?auuocsl lI f [d'g] lurod i(ra^a teql A{oqs ol sjcgns }l

'U Jo ssauPa}ceuuoc aq1 a,rord

ol1[t'0] - I ) ?,(ue ro; oa = (t)"1,(q paugap g-uo qled eq] aq oI p"I'too.t'4

'p?peuu@ fiylul'tspuD pep?uuoc st 'aaoqo pautoldxa sD peptulsuoc 'g acottns ?qJ 'T'Z Btutuarl

'ur uo ernlcnrls

xelduroc parrnba.r aq1 splarf {(!z'!2)},tgurel eql }eql ees err,r')Lodz = 4z turl-les 'gg ur ureurop pelcauuoc fldurrs e s\ dn l€qt qrns (or'on) pooqroqq3rau

al€urprooc e e{sl 'U p [d'C] - gf lurod fue ro;'1ce; u1 'Surddeur ctqdrouroloq

e setuof,aq A * U:1, l€qt os Ar uo alnltnrls xalduroc 3 euueP a,n '1xe11

'deur Eurra,roc e Jo uorlrpuoc aql sagql"s pue U oluo Ur yo Eutddeursnonurluoc € sr l 1eql eas o1 fsea q 1l

(uolllnrlsuof, eql ,cg 'a : (la'gl)r. rq

ua,rr3 uorlceto.rd eql ee U * A:)L p1 'sceds 1ecr8o1odo1 JroPsneg € seuoteq

g ueql 'A u\ 4 Jo spoor{roqq3rau pluaurepunJ Jo uralsfs e ouuep s^\ 'd2 aseql

ig,'h p;" dn uea/$leq aauapuodsauoc euo-ol-euo Isf,Iuouef, e a^€LI e,t 'uteutop

palcauuoc ,tldurrs e sr. d2 acurg 'D o1 d uroq d4 ur pauteluoc qled frerlrqre ue st';2 pw d4 ur lutod e sr b 1eq1 .{eirrr e qcns ul 2I ul lb'bC.9] slurod II3 Jo les aqt

4p fq alouaq 'U ul ul€ruop pelceuuoc fldurrs-e $ qtul^r d lo dn pooqroqq3rau e

e{el 'g Jo ld'Cl

- { lurod {ue rog '9, uo ,(3o1odol e ecnPor?ul ol Peau an '1srrg

's,lrolloJ se U Jo eceJrns Surra,roc l"srelrun e satuoreq B qqt leqt ees eA\'[d'g] sasselc acuale,rrnba

aseq? ileJo les eql eqU lef '(d'C) Jo ssplt acuele,unba eqlld'CJ dq aloueq'g

uo tC ol crdolouroq sr j pue d - d !\Tuapanba erc (d' ,g) pue (d'9) srted oarrl

aseq; 'd o1 od uror; A uo C r{led fue pue U uo d lutod fue 3o rted e eg.(d'C)

te1 'U eosJrns uuetuell{ ualrE e uo od lurod es€q 3 xld 'ecsJJns ulretuelg e Jo

ecsJrns Surre,roc l"srellun e i(lalarauoc lcnrlsuoc lleqs arrr sqled Sutsn fq 'alotr1

A uo CWed e sI Ar uo,, qled e o't11ll v 'd = (ilu tlA u\ il,"r;3:"9:r|"';

61 pre6 sl q!. q d lurod y 'U ecsJrns uueruelg e;o Sutrerroc e eq (g')L'A) p"l

'0),C lurod leururrel eql pue (0)C lutod prltut aq1

qf!^A U uo ,C . C rlled e 1aB aar 'rg;o lurod FIlluI eqt qq^a C;o lutod Ieunurel

eq1 Eurlcauuoc ,(q '(g),9 = (t)C lsq? qcns A ao tC Pue C sqled oarrl rog

s8urraaog l"sra^ru1'Z'Z

30 2. Fricke Space

F1r,"; ; I x I - E b"t*""n i and l lo,polby sett ing F1t,"; = [C1r,,; ,r"(t)] for(r, s) € 1 x 1. Therefore, we conclude that E is simply connected. B

On putting these observations together with the uniformization theorem, weobtain the following theorem.

Theorem 2.2. Fo'r eoerg Riemann surface R, there exisls a uniaersal coaeringsurface R of R, which is biholomorphic to one of the three Riemann surfaces A,C , o r H .

Throughout this section, 6 o universal covering surface E of a Riemannsurface r? we always take the one constructed above.

From the construction of such a universal covering, it is easy to get thefollowing lemma by an argument similar to that used in the case of analyticcontinuation (Ahlfors [A-4], Chapter 8).

Lemma 2.3. (Existence and uniqueness of a lift of a path) For any pathC on R with initial point p, and for ang point F of R oaer p, there erists a uniquelifl C of C wilh initial point fi.

Ttreorem 2.4. (Litt of a mapping) For Riemann surfaces R and S, let(R,Tn,R) and (S,trs, S) be their uniaersal coaerings construcled as etplainedpreaiously, respectiaely. Then giuen an arbitrary continuous mapping f : R- S,there etists a continuous mapping it fr,--- S with forp= osoi. Thit mapping

I is uniquety tletermined untler the condition that i@) - (1, where fu € fr. and

4r e S are such that rs(Q1) = f brn(Fr))Morvooer, if f is differentiable or holomorphic, then f is also differentiable

or holomorphic.

This mappine i t Fl.- ^9 ir called a lift of f: .R * S.

Proof of Theorem 2./. Setting fu = lCr,pr] and 4t = [Dt,/(pr)], we get amapping def ined bv f ( lc ,p l ) _ lDt . f (Ct) - t . f (C) , / (p) l Jor a l l po ints [C,p]in R. Then it is obvious that /(f1) - {1 and fSrp - nsol. Since zrp a.nd n5are locally biholomorphic and / is continuous, / must be continuous. It is alsotrivial that if / is differentiable or holomorphic, then so is /. the uniquenessassertion follows from Lemma 2.3.

Rernark. (Uniq:reness of universal covering) For any two universal coverings(R,r,R) and (r?1 ,11,R) of a Riemann surface r?, there exists a biholomorphicmapping g of R to r?q with Trog - n. See, for example, Ahlfors and Sario [A,-6],Theorem 18A of Chapter L

D

e Jo uorl-rugep ar{} ur uorlrpuo? aql sessrl€s qclq,$ u ur d 3o 2 pooq.roqqStaue esooqo 'ld'Cl = 4'@)"

- d 1as pue 'g f gl tutod € a{€t '(rr) aes o5" '(g),0 - i sagsp, 'l"C) -- L

t€rl? aas am ', IC . 7'C - o5l 3ur11n4 'A uo zg pue IC sqled euos rc1 fd'z7l = p

pue [d'rCj = 4 a,re{ er'r uaql 'd = (!)y = (4)a 1eq1 esoddns '(r) a.,'o.rd oa '{oo.t4

'Q* X u(y)r pqlqcns J ) Lsluauala fiuout QaTgu{Isollt' ?o a.t'o a"r'aqy'g

lo y Tasqns Tcoilutoc fiuo.to!'st IDW:ry uo fipnonu4u@srp fr.1.r,ailo.ti1 sSao i (rrr)'sTurod pat{ ou soy fiyquapt. ayy.tol Tilacxa

1 lo yueu,a1e q)De'"tolncqtod q'{p!} - J ) L fr^raaa "rot Q = nU (dL

?Dql 1l?ns U u! 4!o 2pootl.roqq\nu alqlpns o fl ereql'U ) g f.raaaiog (n)'@)L = D

ltlln J a L \uau.ta1a uo s?srse atayT '(p)tt. = (4)v q?!n U ) !'q fruo .tog o:sa4.tado.td 6utno11ot aq7 sa{stqos g acottns uuvur?tg

o Io (U':r-'A) to .1 dnotf uorTout.tolsuo"tT 6ut.r,eaoc losJeaNutu eqJ 'g'Z BtutrroT

O 'a,rr1calrns sr acuepuodsa.rroc

srql ecueq pu€'.[C] - l, teql seqdrur 7'Z ruaroeqJ'(d'A)ro Jo luetuele u€sr [9] ecurg '(t'd'"tl).[C] = ["d'Cj = (l'd ''I])t reqr s^\oqs g'U €urureT snql'flarrrlcadsar'g;o s1u1od Ieurural pue Ierlrur aql ar€ I'd'Cj pue ['d'oI] pue',

Jo lJll € sl , acuaH'od lurod eseq qlu{ g. uo qled pesol? s sI Col, -

C l€I{}sarldurr y JLov uorleler aql ueql '(l"d'"tl)L ollod'oll uroq Ur rio qled e al QIe.I'J 3 ,L luaurela ,tue a1e1 'a,rr1celrns sr ecuapuodsarJoc slq] 1eq1 aaord o;

'err,r1celur st eeuepuodseJJo? sltl]

l€qt sA^olloJ II'ed'a)rL Jo luatuale lrun eqt q [u ef,uaq Pu€'07 o1 ctdolouroqsr op 'snq;'[t'O] = 1 ) t fue to1 od - (l)'l reqr qrns U uo qled eq] sl o1 ereq!\

'lod' o If = fod'

o Cf = (l"d' " tl).|'

Cla^eq a^r

ueql 'J Jo lueurele lrun eql q -[?] 1eq1 asoddns 'arrrlcefut q ]l ]€q] e,rord oa '.7

o1 (od(g)tv;o rusrqdroruouor{ e sr ecuapuodserroc srq} }tsqt pI^Ir} sl rI'loord

'(A'o'U) 0ut.taaoc losrearun n to 1 dnot'6uotTotu.totsuo.r,T |uuaaoc losraarun ?ql oluo A Io ("d'A)rv dnot6 pTuau,opunt aql

to tustrlilrotuost. uo spyaffi -['d - log)acuapuodsauoc aaoqD eyJ'g'Z rraroaqJ

'(A'v'A) Jo uorleruJoJ

-sue.r1 Sur.rar'oc e sr 1r 'sl

leql '..,1 o1 s3uolaq -[op] srql 'uo1]-ruuep eql fq 'fpea13

ry > la'c) 'ld'c. 'c) = ([d'c1).1'c]fq g uo -[op] uorlce eql eugep e^{'(od'U)rv>['C] ]ueuele fue rog

'f, 1o (od'g)rY

dnorE leluaurepunJ eqt o1 crqd.rourosl sl J dnor3 Eurraaoc lesrelrun slt l€q? aas

II€qs alll 'p ace;rns uueuerg e Jo (g'.u'9,) ece;rns Eutrarroc IesJeAIun uarrrE e rog

sdno.rg uor+BrrrroJsuer;, EurreloC lesrallun'e'Z'Z

ITs8urra,ro3 l"sra^rufl'Z'Z

g2 2. Fricke Space

covering map in $2.1, and denote by U the connected component of r-r(J)containing f. Actually, it is sufficient to take a simply connected domain Uconta in ing e. l t1(0)n0 + { for some 1€ f , then there are points fu, fu e0with f1 = l(it). Since ro7 = n, we get T(fr1) = r(4), and hence Q1

- f i1,for

r is biholomorphic on U. Thus we have l(Ft) = id(Ft), where fd is the identity.By Theorem 2.4, we conclude lhat 1

- id.Finally, to verify (iii), assume that there exists a sequence { Z" }f,r consisting

of mutually distinct elements of l- such that 7"(1() n I{ * / for all n. Thenfor each n, we can take two points [n,Fn € 1{ with fn = .ln([n). Since Kis compact, taking a subsequence if necessary, we may assume that { drl1T=r,{i" }Lr converge to Qo,io € /{, respectively, as n + oo. Since zro7, = ?r, weobtain r(4") = r(f") and o(4") = r(i,). Take a neighborhood [/ of r(,i'") in .Rsatisfyigg the condition of the definition of a covering map in $2.1, and denote byU and, I/ the connected components of zr- 1(U) containing f, and fo, respectively.Since { j"(q") }f-, converges to fo, we hav,e .y"(y)n0

t' g for a sufficient-ly largen. Since ro7"(0) = (J,it follows that 7"(U) = 7, namely, ("tny)-|,1n(0) = 0.By the assertion (ii), we conclude that 7,.11

- 7,. This is a contradiction. !

Exarnple 3. Here is a"n example of a group which does not act properly discontin-uously. Let a be a real number not equal to 2r multiplied by a rational number.Then the group generated by l(z) = edoz does not act properly discontinuouslyo n C - { 0 } .

2.2.4. Representation of Riemann Surfaces as Quotient Spaces

We shall explain a way to construct a Riemann surface Rlf fro a Riemannsurface R and a subgroup l- of the biholomorphic automorphism group Aut(R),where f is assumed to satisfy the properties (ii) and (iii) in Lemma 2.6, thatis, every element of f except for the unit element has no fixed points in E, andacts properly discontinuously on E.

Two points F,C e Rare said tobe f -equiaalentor equiaalent uniler f if thereexists an e-lement .f e f satisfying 4=t@). Denote by [f] the equivalence classof fi. Let R/f be the set of all these equivalence classes p], which is called thequolient space of r? by .i-. Define the projection r'. R * R/f by r(fi) = \fl.

We introduce the quotient topology "n fr,/f . A subset U of R /f is said tobe open if and only if the inverse image o-t (U) of [/ is an open subset of E. thep_rojection r is readily seen to be a continuous mapping of E onto tr/f. Since,R is connected, so is r?/f. Moreover, we see that Rl f is a Hausdorffspace, forl- acts properly discontinuously on .R.

Now, we define a complex structure o" n/ f as follows: for any point f e E,take a neighborhood. Up of I satisfying the property (ii) in Lemma 2.6. We mayassume that there exists a local coordinate zp on 0p. Then, putting p = r(fi),Uo =,tr(0), we see that r: 0O - tJ, is homeomorphic. Hence, setting zo =zpozr-r, we conclude that { (Up,zp)}rrrt l , defines a complex structure so ihut

(z'z)

G'z)'29-r

, -; ,'a - (z)L

, s, u?lprn oslo st slttJ 'T = "lql -

"lrl ,ltp^ ) ) Q(o anqm

,p+z!_k)L9t zo

ru.ro! o soq (V)nV to Tuaurap fi.r'aag (nr)'0*Dql?n C)q'oanym

'q+zo=(z)L

ut.rol o soq (g)wy to Tuaurala tuaag Q1)'I = "q - pD qwn c ) P'c'q'o anaym

,P*zc _k\LQ* zo

ur"rot o soq (g)7ny lo Tuaua1a fr"raag (r)

@'z)

(s'z)

8'Z stutuoT

:sturoJ 3utmo11o;

erll e^sr{ Il pue 'V 'C'Q sul"tuoP l€?Iuoue?;o surstqd.rouolne erqdrouroloqrg

srrrerrroq [BcruorrBC go sdno.rg rusrqdrourolny crt1daoruoloqrg'1p'8'Z

' 1g)tnv pue' (v)tnv' (o)tnv' (q)nysdnor3 ursrqdrourolne crqd.rouroloqlq aq1 fpnls sn 1a1

'puttu ul qql qllft'g, uo flsnonulluo?$p fFadord E?" Pue'fllluepl eql roJ ldacxa g' ur slutod paxg

1noq1r,n slueruela Jo slsrsuo? '19)tnv;o dnor8qns 3 sl J '9'A €Luluel urord 'ur

Josursrqdrourolne-clqdrouroloqtqlo dnorE aql (U)t"V,(q alouap eAyI/ to 'i'Q

seceJrns uuetuerg "".rq1 nq1 jo "uo o1 crqdrorioloqlq q U 'ureroeql uorlezrur.rd;

-run aql Jo anlnl fq 'era11 '.7 dno.r3 uolleuroJsuerl Sfrueaoc l€srollun 8ll ,(q

Ur eceJrns Surra,roc lesralrun e p J/A eceJrns uusruarll luarlonb eqt fq Paluaslerdar sr U eceJlns uu"IuaIU ,(re,ra 16q1 ueas aAeq eal 'uollcas Surpeee.rd eql uI

suorleurroJsuBl,I, snlqgtrAtr'8'U

'@)".--,Wacuapuodsa.u,oc ?Ul repun. A o7 Tuapanba filloatydtotuoto\?q q J nq V k I /Uacottns uuotu?ty Tuat1onb eqt u?qJ 'tr dno.t6 uotyont"totsuntT 0uuaaoc losJearunqnn A acottns uuour?rg o to |uueaoc losrearun o eq (A')L'A) pI 'Z'Z rrraroaq.1,

'uorlresse Surrrrrollo; eq1 ,(lalerpeurul el"q a \ ueql 'J ,(q tf 1o acopns uuouery

TuatTonb eW J /A eteJrns uueruelg sql ilec eM'J /A 3o Euirarroc e st (tr f g';u'g)

suorl"rurolsu?rl snlqol I 't'z

34

w h e r e 0 e R a n d a e A .(iv) Eaery elemenl of Aut(H) has a form.

t(z) =

where a ,b ,c ,d eR wi th ad , - bc = I .

2. Fricke Space

a z + b(2.6)c d + d '

In (2.2), it is sufficient that complex numbers o,, b, c, and d satisfy the con-dition od - b" # 0. However, 7 does not change when a, b, c, and d are multi-plied by a common constant. Hence,-we may normalize the expression of 7 byad - bc - 1. Every element of Aut(e) is called a M1bius transformation or alinear fractional transformalion.In particula.r, an element of Aut(H) is called areal Miibius transformation or a real linear fractional transformalion.

Proof^of Lemma 2.8. First of all, let us determine the form of an element 7 €Aut(C). If 7(oo) = oo, then in a neighborhood of oo, 7 has the Laurent expansion

t Q ) = " , + i b n z - n ,

where a I 0. Then tQ) - oz is holomorphic on e , and hence the maximumprinciple shows that lk) - oz must be a constant function, say 6. Thus we havel @ ) = a z l b w i t h o + 0 . I f z ( o o ) = z o # @ , t h e n s e t t i n g t { z )

- t l Q - z " ) ,we see that both 11 and lpl are elements of Aut(e), and 71o7(m) = oo. Thuswe have fo lQ) = I /QQ) - zo ) = a rz *b r , whe re 0 r ,6 r € C w i th o , * 0 .Therefore, 7 is expressed in the form (2.2).

Next, every element t e Aut(C) is extended to an element of Aut(e) if weput 7(oo) - oo. By the above argument, it is obvious that 7 is represented inthe form (2.3).

Let 7 be an element in Aut(A). Set 7(0) -

B. Then the Mcjbius transfor-mation r(z) = (, - 0)/G - Bz) belongs to Aut(A). Hence .y2 - 1*.r alsobelongs to Aut(A) and 92(0) = 0. Schwarz' lemma implies that 72 is a rotationtrQ) = eiqz,with real number d. Hence, T is expressed in the forrn (2.5). It iseasy to see that 7 is written in the form (2.4).

Finally, for any element 7 e Aut(H), taking a biholomorphic mappin gT(z) =(z-i)/(z+i) of fI onto .4, we have an element lr = ToloT-r e AutlA). Thus 71is a Mijbius transformation and is represented in the form (2.2). Since 7 sends ry'onto itself, we may assume that c, D, c, and d are real numbers, and ad- Dc ) 0.Therefore, this 7 is written in the form (2.5). tr

For more on the fundamental properties of M6bius transformations, such astransformation of circles into circles, and the invariance of the crms ratio underthem, we refer, for instance, to Ahlfors [A-4], $3 of Chapter 3; and Jones andSingerman [A-48], Chapter 2.

Now, for every 7 e Aut(e) given by

'slueruel€?s 3urno11o; eql e^€rl a,r,r 'uor1e1nc1ec aldurs e ,(g'oz - (oz)L Surr(;sr1es C > oz

IIs Jo tas aqt'sl

ler{}1'L1o oz slurod pexgJo las aql eq (t)xrg 1a1 'r(1r1ujpr eq} }ou sr qcrqn

,I = cq - pD ,) ) p,c,q,o ,'.1 1,i = @)LQ*zo

.{q ue,tr3 uor?euroJsuer} snrqory e aq I 1a1

suorlBurroJsuBrl snlqgl tr Jo srrrroJ lBcruouBc 'z'e'z

'{1aar1aadsa.r '(1'1) atnTvufts to dno.r,6 fil,opun lonads eq1plle 7 aa.r6ap lo dno"r6 nautl yonads lDa, eqt are (1 '1)29 pue (U'6)79 araqirr

' { t+} lfi 'r)ns = G't) nsa = (v)wvpue

'{ r+ }/(u'?,hs = (ll'z)tsa = (n)pv

a^eq a^{ 'flrelrurrg'{(e)t"v ) Ll 6oLor-.{ } = (ra)lny reqr pue

uorleuroJsuerl a.,rr1ce forde o1 spuodser.roc (3)tnv Jo (p + zc)l&+ zD) - (z)1, lueurale ue 1eq1 ees e/truaq;,'rd Jo eleulProoc snoeueSouoq e sr [tz : 0z] araqrrr 'rz/02 = (ftz : ozl)g

f,q ? * rd :J Surddeur crqd.louroloqrq e euuep 'paepul 'rd Jo uorleruroJsu€rl

a,rrlcefo.rd e o1 spuodsauoc (3)lny Jo luetuela ue '1d aceds err,rlcalo.rd xalduroc

Ieuorsuaurp-euo aql qlra pagrluapl q C a.raqds uuetuerg eql ueq1\ llrDureq

'(c'dtsul yT sluauala o,rl!,1 ,tq peluasarder sI ,L 1eq1 elop "L Jo uorlD?uesa.tdat rt.tTou epelle)c $st (g)l"y 3 ,L luaruele ue rog (1,)r- W lo V lueuala wtr '4 aa.rfap lo dno.t6toauq lotcads aar,Ttato.td eql tr lpc pue {1+}/6'dlS = (C'Z)lSd tas aIA

'{r+}/6'z)ts = (q)wvursrqdrourosr ue

secnpur ltl (ruaroeq? tusrqdrouoruoq eql dq 'ecua11 'xrr?Bur lrun aql sI 1 ereq^r'{

f + 1 sl W Jo loura{ aql ueqtr, 'eloqe se y Jo saulue eq? er€ p pue 'c'g 'o ereqrrl'(C'Z)1S )> y .ro; (p+zc)/(q+zo) = (z)(y)W tq paugep (q)ny oluo (3'6)79yo ;;4r usrqdroruoruoq e e^eq em'r(lasrarruo)'(C'dlS dnor3 reauq letcads aq1 ;o

lp c1l'^ -l =VLq D )

luauele u3 aAeq e^r

D+z) (1__-i_ - (z\L9*zo

suorl"urrolsu"rJ snrqol{'t'z

' lz:ii';il = li:1, -, | :; l

9t

,I =cq-pp qll^{ j > p'?'q'D

36 2. Fricke Space

(i) The case where oo € Fix(7),i.e., c = 0.lf a/d = 1, that b, c - 6f = *1, then 7 has a sole fixed point oo and it is

written in the form

1 e ) = z * b ,

where b is a non-zero complex number. On the other hand, if a/d f 1, then 7has another fixed point zo, and is represented as

u - z o = \ ( z - z o ) ,

where to =lQ), and ) is a complex number equal neither to 0 nor to 1.

(ii) The case where oo f Fix(7), i.e., c f 0.If (a * d)2 = 4, then 7 has a sole fixed point zo and it is written as

w - % = ; : 1 + o '

where to = lQ), and a is a non-zero complex number. If (c* d)' # 4, then 7has two fixed points z1 and 22, and it is represented in the form

W - Z t , Z - Z l

" , - t r - ^ 7 - '

where w = 1(z), and ,\ is a complex number equal neither to 0 nor to L.

Now, two elements 7r,jz e Aut(A) are said tobe Aut(X)'conjugate or con-jagale in Aut(X) if there exists an element 6 e Aut(X) such that 1z = 6o"ho6-t ,where X is one of C, C, H, or A.

This leads us to the following lemma.

Lemma 2,9. Eoery Mdbius tmnsformation 7(f id) has otue or two f,xed points

on e , antl is Aut(e)-conjugate to the foltowing M|bius tmnsformation 7o:

( i ) I f t h a s a s o l e f i x e i l p o i n t , t h e n T . Q ) = z + d f o r s o m e a € C , a * 0 .

( i i ) I f 7 has two f i ted points, thenTo(z)= ) ,2 for some \ € C,) +0,1.

We call this 7o a canonical form of 7. Matrix representations of canonical

forms in (i) and (ii) correspond to

respectively.A real Mobius transformation 7(l fd) whose fixed points a.re in fl = RU{ m }

is Aut(H)-conjugate to a canonical form 7o such that the entry a or ) of a matrix

representation.To is a real number.

[; ?] , tf ,f.^),

'peuueplla,lr sr (1,)rr1 a.renbs sll Tnq '1, fq flanbrun peurrurelap1ou sr (l).r1 snql'(p+D)- olw peretle sr ec€r1 slr ueql'V- f,q pacelda.r sr y

;1 '.{. uorleurroJsusrl snlqgl4l e Jo nD4 " pelpc q qcg/'r 'p * o - (l)r1 1nd a14

'T = ?q - PD ,C ) p,c,q,o ,l\ '^l =,L9 D J

:,L;o uorleluasa,rdar xrrleru " Jo ecerl eqlnp+p tsql atoN 'Z+y/I+y=

e,(p ar) uorlenbe aql segslles y raqdrlptu s11'ol,;o

lurod pexg e^r1?€r11e a{} o1 Eurpuodsarroc,L;o lurod pexg sJo alloq? aqluo spuadep yy/I pue 1''a'r'l,yo.rar1dr11ruu aqt roJ sarroqr o,lrl e eq a64'l;o.taqd41nut, e pallsc sl y slql'0'O* y'C ) y) zy -

Q)'L uroJ Iscruouer € otapSnluoc-(g)tnv o. cqoqered lou sr rl?rqar ,L uorleurroysrrerl snrqotr tr e 'alo1q

'g uo sTurod pae{ omy soq L ta fipo puo tt, ctloqtedfitl s! ,t (H).zz = rz puV ,

*H) zz 'H ) rz pUl qtns zz 'rz slutod par{ on\ sorl L tt fi1uo puo tt, ctTdqla st' L

'g uo Tutod pat{ alos o soy L lr fiyuo puo lt cqoqn.r,od sr L

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eql lou sr q?rq^r'(V)lnV ro (11)7ny Jo luetuale fra^a 1€ql ^\oqs ol fsea st 11'fla,rrlredsar '1,

go lurod pexg a^rlrer11e a{l pue lurod paxg Suqledar eq} !o pue r; ,(q alouaq'flarrleadser 'L

1o yut,od par{ aatyco.tqqo elql pve Turotl pax{ 0ur11ada, eq} pell€,elre ez pue rz uaqJ, 'I < lVl qll^ y euos to! zY -

Q)'L ruroJ l"tluouec e Jooo pu" 6 qurod pexg eql o1 puodsarroa 'fla,rrlcadser'zz pu€ Iz 1eq1 asoddng'l, uorleur.ro;suer? snrqoq cnuorpoxol e;o slurod pexg aql aQ zz pue rz lo.l

'seuo ?rloqredr(q apnlcur suorleurroJsusrl c[uoJpoxol eql

l€ql eurrrnss€ airr '4ooq slql uI 'crldqe rou ttloqersd .raqltau sl ?l JI cruoJporoleq ol pres sr flrluepr eql lou s! qrlqa uorleurroJsuerl snrqory y'((oo'O] / Vpu€ I I lVl'C > f) zV : Q)L.{q ue,rrE sr aldurexa uy 'cqoq.redfq rou 'cr1dq1a'cqoqered raq?reu are qf,rqra suorl€ruroJsu"rl snlqol I are eraq+ teql aloN

'I+y '0 < yeruosroJ zy = (z)oL uortellpeolale3nfuocsrlI;r ctloy,adfiq qf (U)'(7 3 u) ttuT

* 0'1g )f g auos tol zsp - (r)'L uorlelor e o1 ele3nluoc sr ]I ycr,Ttlqla q t (I)'o+n'c>p

eurrros roJ a + z - (z)ot uorlelsuerl e o1 ele3ntuoa s! 1l y cqoqotod q f (t)

'f1r1uapr eql tou $ qcrrlAr uorl€ruJoJ-suerl snrqontr e eq ,L 1a1

'sadf1 earql otq suorl"ruroJsusrl snlqotr tr fSsselc all

suorlBrrrroJsrrB.LT. snlqg,tr tr Jo uorlBcgrssBlc '8,'8'z

(r)(r)

LTsuorl"urolsuerl snlqgl I 't'z

38 2. Fricke Space

By a simple calculation, we see that Mobius transformations are classified bytrace.

Lernma 2.LL. Let 7 be a M6bius transfonnation which is not the idenlity. Thenthe following hold:(i) 7 is parabolic if and only if tf (7) = a.

(ii) 7 is elliptic if and only if 0 f tf (1) < a.(iii) r is hyperbolic if and only if tf Q) > .(iu) r is lorodromic if and only if tf (1) e C - [0,4].

Finally, we define the axis of a hyperbolic real Mobius transformation 7.Suppose that 7 is conjugate to a canonical form U@)

- )z with ,\ > 1, by areal Mcibius transformation 6. Namely, suppose that 7 = 6oloo6-1. The half-lineL = {iy | 0 < y < oo} in the upper half-plane .Il is the geodesic, joining 0 andoo, with respect to the Poincar6 metric ldzl2 I [m z)2 on I1 (see $ 3 of Chapter 3) .The image 6(L) of .L under 6 is called the oris of 7 and is denoted by Ar. Then,4', is the geodesic joining the fixed points r,, and a., of 1, which is characterizedas a semi-circle which joins r., and o., and is orthogonal to the real axis. Similarly,we define the axis A, of a hyperbolic transformation 7 in Aut(A).

2.4. Fuchsian Models

First, we show that a Riemann surface whose universal covering surface is notbiholomorphic to the upper halfplane f/ is biholomorphic to one of e , C, C -

{ 0 }, or tori. Next, we study some fundamental properties of discrete subgroupsof Aut(H), i.e., Fuchsian groups.

2.4.L. Riernann Surfaces of Exceptional Type

Let us determine Riemann surfaces whose universal covering surfaces are biholo-morphic to either 0 or C.

Theorem 2.L2. A Riemann surface^ R has a uniuersal coaering surface fr. bi-holomorphic to lhe Riemann sphere C if and only if R itself is biholomorphic toC .

Prool. Assume that .E = e . Sitt"" every element 7 of its covering transformationgroup f is a M<ibius transformation, it should have fixed points. Flom Lemma2.6,any 1e f - { r .d} has nof ixed points ot t e . H" t t .e we have | = { id} , whichimplies that .R = R/f = C.

Conversely, suppose that ,R = e . Sitt.u e is simply connected, by the con-struction of E in $2.2 it follows that E = e . tr

Theorern 2.L3. A Riemann surface R has a uniaersal couering surface biholo-morphic to lhe complex plane C if and only if R is biholonrorphic lo one of C,C - { 0 } , o r t o r i .

tr 'zry = @)rL ^q palereua3 sl J leql qcns Iy raqrunua.Lrlrsod e slsrxe a.req1 'g uo flsnonurluorsrp dl.radord s+re J ecurs .y reqrunuaarllsod auos roJ zy - (z)L yr fpo pue yr o,L qlyr e^rlelnuruoc fl (g)7ny ) Llueuala ue leql ^{oqs ol fsea sr ,1 .19)WV ur uorle3nfuoc ,tq (g < oy) zoy- (r)"L leql aunss€ {etu a,u uaql 'orToqred{q sr oL Wql asoddns ,,no11

'rq + z = (z)rL fq pale.reua3 sl J teqt q)ns rg requrnua,rrlrsod e s?srxa araql 'g uo dlsnonurluoc$p dl.radord slce J a?urs .tI ) g auros.rog g * z = (z)L urroJ aqt ur uellrrA\ sl f JI ,,l1uo pue gr o,L

{tgl a^rl"lmuurot* (g)wV 3 l" luaurala u€ leqt ees ol fsea sr 14'(g)nV ur uorle3nfuoc fq

@*'q'U > og) oq+z - (z)oLryq1 erunsse feur ear uaql ,cqoqered s o,L;1'cqoqrad{q

.ro crloqered sr o,L 1eq1 saqdurr 0I'Z etuure1 'H uo slurod pexg ou seq ol, acurg'p!

* oL qtl,lr J 3 o,L luaurale ue a{€tr '{pgl1

* J leqt arunsss,(eu e11 .too.t4

'ct1cfic s, lN u?Vl 'uoqaqo s? J II 'H uo snonut?uocsrp

fr4.rado.rd s! J to uorl?o eW llUI q?ns puv g uo sTutoil pac{ ou sly {p?} - Jlo Tuaua1e fi.raaa Toqg qcns (g)nv {o dnolfiqns D eq J pI .VTZ BururaT

'adfi7 TouorTdnr? lo eq ol pres sr

rrol ro '{ O } - C

'C 'C Jo euo o1 erqd.rouroloqlq q qcrq^\ ac€Jrns uueuer}I V

' .t / C ot cryd.t out oloqrq

s! A pW q?ns J dno.tf ac47ol D slstr,? a.r,aq1 'g sn"to7 fr.tana "lo3r .z(.re11o.ro3

D '3 o1 crqd.rouroloqlq sr snrol € Jo aceJJns3ur.ra.,loc I€srelrun e e)ueH 'dno.r3 erlc{c e aq pFoqs J e qcns leq} slras$ qcrrl^r(p1'6 eurtual) eurural 3ur,rlo11og eqt s1?rp€rluoc srqJ .g.

;o dno.r3 FluauepunJeq? o1 crqdrourosr sl J roJ (6

4uer;o dno.r3 uerlaqe aarJ € eq lsnur J uar{} ,.Fl

o1 ctqdrouroloqlq q U JI'snrol e q g 1eq1 asoddns'fleurg'C = U leqt ^rou{a,$'I'Z$yo 1 eldurefg ur uaes seAr sV.{O} - C =A 1a1,}xaN .C =A 1aB am'palcauuoc fldurrs fl C ecws 'C =A ?eql aurnsse lsrg,asra,ruoc aq1 lroils o1

'flalrlcadsa.r 'snro1 e pue '{ 0 } - C

'9 o1 crqdrouroloq-lq sl J/C -

U ac€Jrns uueuerg eql '(II) pue '(ll) ,(r) sasec ur ,e.ro;araqa

'U re^o luapuedepur fpeauq er€ qcrq^r

C f rg'og autos roy tg* z = (z)rL prrs og *z = (z)ol araqar'(rt'.t) = J (lll)'{o}-c)oq

oruos roJ oq*z - (z)ol uorlelsuerl e {q pele.reueE rl J (sr

leq} ,('tl --,t

{ptl = .t

:(1'deq3 Jo Z$ '[t-V] sroJlqy ';c) sesec aerql 3ur,rao11og

aql rncco ereql l"ql elo.rd uec e^{ ueql '(9'6 eruuel ;c) g uo flsnonurluocsrpfpadord slc€ J teql [eeer em '.ra,roaro1,1i 'g uo slurod pexg ou seq {pl } -,f > ffue asnereq'l = D'raqlrng'(O f "'C f g'o) q + zo - (z)L turoJ eql uruellrr^,l\ sl J ;l,L fre,re g'A"uure.I fq,(g)lnV;o dnorEqnse sl J acurg.dno.rEuol+€ruroJsue.rl Sur.ra,roc l€sralrun st! eq J lel

'C = U leqt aunssy '{oot4

(tr)(r)

6tsIePoI{ u"rsrlf,nJ 't'z

2. Fricke Space

2.4.2. Fuchsian Models and f\rndamental Dornains

The following is an immediate consequence of Theorems 2.I2 and 2.13.

Theorem 2.L5. A Riemann surface R has a unioersal couering surface fr, bi-holornorphic to H if and only if R is not ̂ of exceptional lype; that is, if and only

i f R is not b iholomorphic lo any one of C, C, C - {0} , or tor i .

If a universal covering surface E of a Riemann surface ,R is the upper half-plane fI, we call its universal covering transformation group I a Fuchsian rnodel

of .R. In this case, f is asubgroup of Aut(H). However, identifying I/ with 4,we sometimes consider a F\chsian model f as a subgroup of Aut(A).

Remark -1. By an argument similar to that in the proofs of Theorem 2.13 and

Lemma 2.I4, we see that the fundamental group of a Riemann surface R is

commutative if and only if .R is biholomorphic to one of C, C, C - { 0 }, tori,

the uni t d isk .4, 4- {0} , or annul i {z e C | 1 < lz l < r } .

In order to obtain a geometric image of correspondence between a Riemannsurface R and its Fuchsian model f, we use a fundamental domain for f. An

open set F of the upper half-plane 11 is a fundamenlal domain for f if F satisfies

the following three conditions:

(i) z({) oF = / for every 7 e f with 1 { id.

(ii) If .F is the closure of .F in 11, then

a = [J ,r(F).7 e l

(iii) The relative boundary 0F of F in H has measure zero with respect to the

twodimensional Lebesgue measure.

These conditions tell us that the Riemann surface R = H /f is considered as

F with points on dF identified under the covering group l..

Emmple y'. For each covering group l- in Example 2 in $2.1, we define similarly its

fundamental domain. The following (i)", ..., (t)" give examples of fundamental

domains for covering groups of (i)', . . . , (t)' in Example 2, respectively'

( i ) " r - { , e C | 0 < I m z < 2 r } .( i i ) " r - { , e H | 0 < R e z < 1 } .( i i i ) " F - { , e c - { 0 } | 0 < a r s z < 2 r / n } .( i v ) " . F - { z e H l I < l r l < f } .( u ) " F - { , e C l r - a } b r , 0 < a < 1 , 0 < D < 1 } .

There is a simple way to construct canonically a fundamental domain for a

Fuchsian model of a Riemann surface r?. First, cut -R along suitable smooth paths

on ,R to get a simply connected domain Ro. Let .F be a connected component of

aa.rq1 3uo1e U ln? pue U 3 od lurod e e{e; 'tO pue 'zO 're self,rrc ea.rq1 fqpepunoq 'aue1d xalduroe eql ut Ar ureuop e 'g'Z '3ld ur pal€rlsnll se 'raprsuo3

'8'Z'8tJ

'g alilu,oag

'z'z'Btr

€-)t

'6, u1 leeduroc f1a,rr1e1a.r $ qcrqrr ureluop Fluarrr€punJ e ser{ J l"qlsmolloJ l.I 'rslncur€d ul'?,'7,'3U ul patertsnll sB J roJ ureuop le]uetuepunJ esl ('U)r-l;o g luauoduoc pelf,euuoc V'oU ur€ruop pelrauuoe rtldurrse 1e3 aa,r'lg pue fy 1ye Euop g,3ur11ng'od lurod aseq qlr&t salrn? pesolc alduns qloorus

il€ arp {g pue fy trqt qans (6 l) d 6nua3 Jo Ur eceJrns uueruarg pasop " roJ(od,A)to Jo srol"reuaS;o ualsfs l"cruouec " nq,=f{ lg'lV } tet

'g a\duorg

'saldurexa aa,rq1 Eur.nolloJ eql u-r"lqo aal 'fe,u, slql uI'J IoJ uleuroP

IelueuepunJ e sI d slql 1eq1 flrsea aes eAr'Z'Z$ ul (A'y'A) Eurraloc lesJe^Iun eJouorl?nrlsuoc eql ,tg'.u detu Sur.ranoc eql rapun tA p (;A)t-! a3eurr esra^ur eql

'V

I

TVsIePoI I u"rsqf,nJ 't'z

42 2. Fricke Space

smooth curves Ct, Cz, and C3. By the same argument as that in example 6,we have a fundamental domain tr'for .R as is shown in Fig. 2.3. The elements

Ir,''lz € l- corresponding to the elements [At],[Ar] E q(R,po), respectively, givea ca.nonical system of generators of f. In $1.5 of Chapter 3, we shall describeanother way of cutting .R to get a fundamental domain for this group.

Example 7. As a limiting case of Example 6, let each circle D; degenerate to asingle point pi to obtain a Riemann surface,R biholomorphic to C-{n,pz,ps}.A Fuchsian model of .R is conjugate to the principal congraence subgroup f(2)of leoel 2, which consists of all elements 7(z)

- (o, + b)/(cz * d) such thata , b , c , d € Z , a d - b c = l , a n d a = d = 1 , 6 : c = 0 m o d 2 . A s a s y s t e m o fgenerators of f(2), we have fQ) = z *2 and fQ) = z/(22 * 1). The pictureon the left hand side of Fig.2.4 shows an example of a fundamental domain forf(2). The picture on the right hand side of this figure illustrates a fundamentaldomain for a subgroup of Aut(A) conjugate to l-(2). For details, see Ahlfols

[A-4], $2 of Chapter 7; and Jones and Singerman [A-48], Chpater 6.

----)z - t

R

rQ)

I

Fig.2.4.

Remark 2. As canonical fundamental domains for a subgroup of Aut(H) actingproperly discontinuously on ff , we have Dirichlet regions and Ford regions. Fordetails, we refer to standard text books such as Beardon [A-11], Ford [A-31],Jones and Singerman [A-48], Lehner [A-66], [A-67], and Maskit [A-71].

a fundamental domain in .F/ for a fundamental domain in 4 for f(2)

H ) on <- (oz)uL leql q?ns J Jo sluatuela lcurlsrp Jo I;:{ 'l } ecuenbas e pu€

H ) oz lutod e eAeq e^l ueql 'I1 uo dlsnonurluocsrp dpedord lce lou saop Jleql aurnsse 'f1asra,r,uo3 'uor?rugap eqt ,tq snor^qo sr (r) seqdu4 (ll) teql /oota

'g uo filsnonur?u@stp filtadaul sTco'uDtsq)nf, st

:Tuapatnba a.ro 6utmo11ot ayq (g)lny to tr ilno"t\qns o ro4 .lI.U uraroaql

D 'a1e1duoc fqaraq sl gI'Z eurruerl ;o;oord eql '(I) saqdrur(r) ecuag 'elercsrp

tou sr J '; go lurod palelofl ue lou sl J Jo p? luetuela lrun

eql ecws pue 'u fue .ro; pg iL r+uLor_("f) p* J J I+ul,or_(u,[) acurg .Q1)lnV

ul r-f o1 saS.rerluoc tfi{ ,_("t) } W{t aes e,ra 'suorleluase.rda.r xrrleur neql 3ur

-raplsuoC '(U)t"V I ,L luaurela ue o1 saS.raauos r{rrq^\ J Jo sluauela ?cultslpJo I=":{ ul,} acuanbas ? s}srxa ararl} leq} arunss? '/'roN '(l) serldurr (rr) ecuaqpue 'p1oq

lou seop (rr) uorl.rasse eql ueql 'uersqcng lou sr J y'f1tea1a 'too.t4

'(11)wY uPefitaouoc Wyln J to sTuau,ep purlstp fi11onynu.t to sacuanbas ou ?srse ereqJ

'dno.t6 uvrsqcnf, o s! J

:Tuaparnba ato 6utmo71ot aq7 (g)ny lo ,7 dnol0qns D ro4 .gT.Z BrrruraT

'dno.t6 uvrsq?ng e se ol perraJer osle fl (V)l"V l"dnor3qns elercsrp V

'(C)lnV rc'(3)wV '(V)l"V;o dnol3qns elercsrp € augepeira'f1.re1turg'sluaurala jo Jaqunu elqelunoc e lsoru leJo slsrsuoc dnorE u€rsr1rr\fe '{1tlqe1unoe

Jo tuorxe puoees eql segsr}es (U'A)ZS ecurg 'dno.r,6 uvrsycnl epalp? q (H)InV;o dnorEqns eterf,srp O

'19)tnV Jo leql uorJ pernpu! J uo

f8o1odo1 e^rl"ler aql ol lcadsar qlrar slurod pelsloflJo slsrsuoc J ''a'l'Gt)lnv

Jo lasqns alansrp e s.t J ! ?pr?erp eq ol pres sr. (11)7ny ;o .7 dnol3qns y'oo ol sPuel u se

'dle,rrlcedsar 'p pr. '" 'q'o o1 aE.rarruor tp pue 'uc 'ug' ro yr fluo pue JI (U 'Z)I S rrl

fl il=,ol saEre,ruoc

l"p "rl -tt "ol= o

qll,!{ (U,Z)IS lo t;3{ "y } ecuenbas aq1 ,ara11 .(U,Z)ZS;o f3olodol

eqt,tq paenpur sl (U'Z)ZSa;o fEolodol aql'I't$ ur ux\oqs $ sp uorl€rgrluepreql repun (1g'Z)lSa dnor3 ar1 eqtJo euo eql 01 luale,rrnba sr f3o1odo1 slql'oool spual u * H Jo slesqns lceduroc uo I o1 .{pr.ro;run set.raauoc "L

I (n)pVI I ol saEre,ruoc 1g)lnV Jo I=J{ u,L

} acuenbas e teql sueeru srql '{to1odo1

uado-lcedruoc eql ''e'r.'(11)WV uo ,(Eo1odo1 Iernleu e Surugap q1ral ur3aq e11

(n)pV go sdnorEqns alarcsrq 'g'V'Z

J (II).r (l)

(s)(r)

tsIaPoI I u"rsqsnd 't'z

44 2. Fricke Space

as n --+ oo. Since { r" }Lr is a normal family, taking a subsequence, if necessary,

we may assume that { 7, }p, converges uniformly on compact subsets of f/ to a

holomorphic function 7 defined in 11 . Flom the following lemma (Lemma 2.18),

this 7 must be an element of Aut(H). Hence by Lemma 2.16, f is not F\rchsian,

and hence (i) implies (ii). D

Remark -/. For a subgroup f of Aut(d), the discreteness of l- does not always

imply that it acts properly discontinuously on e . A typical example is given by

( a z * b. = t t Q ) = * + d a , b , c , d . e Z + i Z \ .

Lemma 2.L8. Let { f" }Pr be a sequence of Aut(H) which conaerges uniformly

on compact subsels of H to a holomorphic function f defined in H . Here, fadmits a constant function with aalue a. Then either one of the following holds:

(i) / is an element of Aut(H).(ii) / is a constant function c with c € R.

Proof. We consider the unit disk 4 instead of I/. Clearly, we have l/l S 1 on

A.If lf(2")l = l for some point zo € A, the maximum principle implies that /is a constant function. Thus either l/l < I on 4, or / is a constant function c

wi th lc l = 1. I f l / l < 1on ^4, then / belongs to Aut(A) . In fact , { ( r " ) - t harbeing a normal family, taking a subsequence, if necessary, we may assume that

it converges uniformly on compact subsets of 4 to a holomorphic function g

defined in 4. In pa.rticular, we have gof = fd. By the same argument, we see

that l9 l ( 1on 4 and f og = id. Hence, / belongs to Aut(H). D

Proposition 2.19. Let f be a Fuchsian model of a closed Riemann surface ofgenus g22. For an arb i t rary point ( € R- RU{m}, therc edsts a sequence

{ f" }Lpppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppr of f such that {6Q,)}f=t conaerges to ( for any point zo € H.

Proof. Take a fundamental domain F for f such that the closure F of F is

compact in I/ (Example 5 in $a.2). By the definition of a fundamental domain,we can pick a sequence {Zr}Lr of l- and a sequence {rr}f;=, of points in F

such that {l.Q^)}[, converges to (. Since {' l '}Lr is a normal family, we

may suppose that {f" }Lt converges uniformly on compacts subsets of ff to a

holomorphic function f defined in 11. Thus Lemma 2.18 shows that / must be

a constant (.

Remark 2.Let L(f) be the set of all accumulation points of the set I lQ")lt el-], where zo is a point on f/. Lemma 2.18 implies that I(f) is independentof the choice of zo.It is well known that I(f) is a closed subset of fr.. We call

I(f) the limit set of a Fuchsian group ,l-. iroposition 2.19 tells us ,(l-) = Rprovided that ft = H/f is compact.

uJoJ oql ur uellrr^\ sl "y lsql aas elt 'u raEalur errrlrsod ,(ue rog

,_(V)oV"V - rluV pus y - Iy Eurllnd 'ol,;o uorleluasardar xrrlpru e q qclqa

' f] 9l ='n [I II

tes e1l 'I > lal > 0 qll,lr (1g'Z)IS 3 y uorleluasardarK:<rrlsru e seq I leql qcns J ) L lueuala u" slsrxa eJeql 1eql aurnssy 3[oo.l4'

'0 # c to1t paprao.ul t ? l"l sa{nqos

| - ?g - pD 1g> p'c'q'o 'l'^ '^l = ,Lq D)

uotToTuasa.til?J ruloru o 6unoq ,! ) L frtaaa u?qJ 'l a z - (z)oL uotyo1-suDrln 0ututoyuoc dnafi uotsycri"l o ?q J pI (lt1?,] nzturrqg) .IZ.Z BrrruraT

tr 'uorlrrpsrluoc € ol speel qcrqal'alarcsrp lou sr J'91'6 eurural ,(q 'aro;a.raqa 'sluetuale

l?qlqp Jo slsrsuor --?{ ? } leql eas e,rl

'l +ypue'0 < y'0 f goacurg'oo- ts uw Cts ?uaql,I < yJI.oo+ ts use

I r ol -I'qt' ; j ='

o1 sa3ra.iluoc "t ueqt'I > y > g y'acua11

I I o'l ' |.(,r, - I)eD ;i

='

1eB ea,r'u.ra3a1ur arrrlrsod {ue.ro; u-VrBuVB - ug 3ur11as'arrolq'flalrlcadsar

,o * qe ,u ) 9,o , [,;,

g]

,r+y ,o<y ,l';. i]

-g

_V

dq uerrr3 an g'L lo g'V suorle?uas-e.rda.r xr.rleur ueql '{ oo } = (g)*1.{ U (l)xrg pue { m'O } = (l)*tg t€rll eunssefeur aar 'uorle3nfuoc-(Hhnv fg 'sp1oq (rr) rou (r) raqlrau l€ql eunssv 'loo.r,4

.0 = (g)xr.{ u (r)4.{ (r)

:sp1oy furmoilo! eyr lo euo t"rl, 'plTg"r;r'::r:;"::,s? L lI 'tr dno.r0 uorsq?nf, o to sTuauale onl ?q g puo L pI 'OZ'Z BruuraT

'ralel pesn ar€ rl?rq,a,r, 'sdnor3 u"rsqcr\{ ;o sarlradord auros luesard all

sdno.rg uBrsqcr\{ ;o sarl.radord Jaq}JqlI'V'?'Z

9tsIePoI{ u"rsqf,nJ 't'z

46

o - l o n 6 " . l" " - l " n d n J '

where o, = I - an-rcn-r, bn = (an-!)2 , cn = -(cn-1)2, and d,. =

Thus it follows that cr = -"'n-t * 0 as n * oo. Next,

max{ lol,l/(L - lcl) }, we obtain inductively lo"l S M for anyant bn, and d, converges to 1 as n + oo. Hence, .Ar converges

contradicts the discreteness of l-.

2. Fricke Space

I * a n - ( n - t .

setting M =

n. Thus eachto Ao, which

tr

Theorem 2.22. Eaery element of a Fuchsian model of a closed Riemann surface

of genus S (|=2) consisls only of the identity and hyperbolic elements.

Proof. Since every element 7 e f - { id } has no fixed points on I/, it is parabolic

or hyperbolic. Assume that f contains a pa"rabolic element lo.By Aut(H)-

conjugation, we may suppose lhatT"Q) - z*1. From Lemma2.20, any element.y (+ id) of .i- with f(m) = oo is parabolic, which is written in the form

ilz) = z *b for some real number 6. Hence, ]-- = { j e f I r(*) = oo} is a

cyclic group. Replacing 7, with another element, if necessary' we may assume

that 7o is a generator for f t . S ince every 7(z) - (az *b) l@z*d) , ad-bc= L,

belonging to l- - ,i-- satisfies c + 0, Lemma 2.21 shows that lcl 2 1. Thus we

obtain1

rmTQ) S 1r-;pp 51

for all z with Imz ) 1. Set Uo - {z e H llmz > 2}. Then any two distinct

points on [/o are not equivalent under any element of f - l--. Thus the quotient

space Do = Uo/f* is biholomorphic to a domain Ro in r?. since 7o corresponds

to a non-trivial element of the fundamental group of .r?, the closure E of R,

in R is not simply connected. Since Do is biholomorphic to the punctured disk

{z e C | 0 < l " l ( 1} , we infer that 4must be homeomorphic to {z € C | 0 <

ltl S | ). This contradicts that R is compact. tr

Remark. This theorem is also obtained by using the hyperbolic geometry dis-

cussed in $1 of Chapter 3. We present its outl ine. Let dsz = ldzl2l(Imz)2 be

the Poincar6 metric on 11, which induces the hyperbolic metric on R - H/f .

Assume that f has a translation 7o(z) - z * 1. For any positive number o,

denote by C" u closed path on ,R which is the image of the segment tro joining fo

and 1o(ia) by the projection r: H -.R. Let l(C")be the hyperbolic length of

Co,i.e., the length of .Lo with respect to the Poincar6 metric. Then we see that

t(C") - 0 as n + oo. On the other hand, .R being compact, we have a sequence

{ o" }L[r of positive numbers such that dn + oo and r(l'o") + po as r, + oor

where po is a point on rt. Hence, if we take a simply connected domain u which

contains po, then the closed path C,. is included in [/ for sufficiently large n.

This implies lo - id, a contradiction.

-,;,j "ii 1.j.,,i,",,; i";:;"Jn ,, u",u^ ,H or { ro / Utt V '/G ol tuale^rnba sr (,<)Y r"qr q?ns /U * A : ! Surdderu crqdlouoloqrq

€ slsrxe ereqt ueqtr '6Jul[g',A) = [3',Ur] l€ql qcns lr uo,3 Bur4.retu € pue6 snua3 Jo /Ar ec€Jrns uueruarg pesolc raqloue e{€I .6A.A uoxllsoilo.t4 lo !oo.t,4

'rldorloolgoln = lld 'fo] eraqm

r=!'P? = lld 't"]L[

u uorrelaJ

Ieluau"punJ alos eq? sagsrl€s qcrq.lr ,s.ro1o"raua6 to uta7sfr,s IoJ.uouDJ slr se ol

parreJar rl I=;{ ld ' lo } s.roleraua3 yo ualsfs eql .[3' ,g] acey.rns uueruerll pesollpe{r€ru e Io pporu uorsqrnl peztlDturou eq1 .7 dno.r3 u€rsqrnd srq} ilec e11\

fi.anbrun s? K ot Tcailsa.r-q7rm suortzpuor ,"n frriJE":Hti:'{:ri:::r"'r:;'Y;::ppolu uvzstlcng o lo

t=ou[!d'ln] s.tolo.tauaf lo ua7sfrs lD?tuouD) p ,(27)6 snuaflo g acottns uuDuery pesop D uo g 6utt1.tou, uant6 D rol .gZ.Z uorlrsodo.r4

'(tt) pun (r) suorlrpuocuorlezrl€rurou aq1 f;sr1es ud pun to lr.I? etunsse .r[eur a,ll ,r(ressaceu y,(g)Wy ul..,ir Surle3nfuor'.raq1.rng 'Q = (6d)xllU(to)xrg seqdrur 0U.Z €unueT ,arrrlelmuuroc

lou ere ud pun to acurg 'cr1oq.red,tq arc 6 j pue to q1oq,Z7-(,tuaroeqtr dq ,1ceg u1'suotllpuoc uor?ezrlsruJou aql segsrlss rIJlqA\ u Jo lepo{u u"rsq)nJ 3 s}srxe sferrrlearaql 'f snuaS;o a)eJrns uu€urerg pesop e uo 3' 3ur>1reu uarrr3 e Jod .tlJDurey

'1 1e lurod paxg a^rlcerl?e sll ser{ t^a (I)

',{1anr1cadsa.r 'oo pue 6 1e slurod pexg a^r}ceJ}le pue 3ur11eda: s1 seq td (l)

:suorlrpuoc uorlezrleurou aq1 esodurl e,!\ ,U uo 3 3ur4.reur ue,rr3 e o1 J Ieporuuersqrnd e ,,{lanbrun u8rsse o} rapro uI .ile,lr se Ur aures eqt Jo lapour uersqrnde sl r_9J9

- ,.7 dnor3 eq1 '(g)wv ) 9 fue ro; ,s1

ryql :(11)7ny go sursrqd-rouoln€ .reuur {q pasnec ,{lrn8rqure egl seq A Io J Iapour uersqcnd e ,lop

'6'"''Z'I = ! qcee roJ 'f1a,rr1cedse.r ,(od,A)to

q [fg] pue [fy] o1 Surpuodsarroc J Jo stueuela oq1 ld pue fo fq alouap ,9.6

utrreroeql ur pelels A lo J Iepou u€tsqcnd e pue (od,U)Iz uae.nleq ursrqd.rourosraql repun 'f snuaS;o U e)eJJns uuetuerg pesol) e go (oa,g') t.u dnor3 plueur€punJaql Jo srolerauaS;o uralsds Ierruouec q ''e'l 'U uo 3urr1.reu n rl t=f{

[lS],[!V]] - 3' eraq^r 'f snuaE Jo [3''U] sec"Jrns uu€urarg pesol) pa{retu II€ }o s}srsuo)

kZ) 0 snuaS;o tg aceds rellnuqrrel eqt ,I raldeq3 Jo t$ ur peugap se^.r sv'QZ) 0 snue3 go

se)eJrns uueuarg pesolc Jo r; eceds refintuq)rel eql uo sel€urprooc e{]r.U pall€c-os eugap fteqs ellr 'slapour u€rsqcr\{ Jo sroleraueS;o ue1s.{s l€)ruorr€c e Bursn fg

oreds a{rl{,{ '9'U

L?acedg arpug '9'7

48 2. Fricke Space

where {ot;,01 }f=, it the canonical system of generators of a Fuchsian model of

.R' which iatisfiLs the normalization conditions with respect to ^D'. From condi-

tion (i), we have iQ) = )z for some l ) 0. Further, by condition (ii), a, and

a! have the common fixed point at L, and hence ) = 1, i.e, i = id. Thus we get

ai = ati and Bi -

Fi. tr

Lemma 2.24. Let {oi,gi}o;=, b" the canonical system of generators of the

normalized Fachsian model i for a point lR, El in To. A an element t(z) =

(az + b)/(cz + d) of {o;, | i}t=, do"t not coincide with Bo, then bc } 0.

Proof. ln the case where 6 = c = 0, we have Fix(7) = Fix(Be) - {0,*},

and hence 7 and Bn are commutative, a contradiction. Next, in the case where

6 = 0 and c * Q,we get Fix(7) = Fix(Be) = {0}. Thus, 7 and Bo being non-

commutative, Lemma 2.20 implies that f is not F\rchsian. Hence we have a

contradiction. By the same argument, in the case where b I 0 and c = 0, we

obtain a contradiction. tr

By this lemma, the canonical system loi,gi )f=t of generators of the nor-

malized Fuchsian model l- for a point [.R, E] in To is written uniquely in the

form

*, = T#,

ai di, ci € R, ci ) 0, aidi - bici = I,

atrz * b,,9i =

ffa 6i, o' i,our,C, eF., " ' i ) 0' dil i - f iCi = |

f o r e a c h j = 1 , i , . . . , 0 - ,

Now, we define the Fricke coorilinates Fo:Tn -* R6e-6 by

f o ( [R , l ] ) = (41 , c1 , d1 , a l , c \ , d1 , . . . , a s - | t c e - t , d g - r , a ' o -1 , c ' n - r , d ' s - r ) .

The image Fn = fo(To) is called the Fricke space of closed Riemann surfaces

of genus 9. The topology of Fo is introduced by the relative topology of Fo in

R6c-0. In $2 of Chapter 5, we shall verify that.F, is asimply connected domain

in R6g-0. By the following theorem (Theorem 2.25), Fs is a bijective mapping

of ?o to Fo. Hence we define a topology on Q by identifying ?, with -Q under

d. Therefore, a topology of the Teichmiiller space ?(.R) of a closed Riemann

surface -R of genus g is induced from that of ?n. In the rest of this book, we

assume that Tn and "(.R) are equipped with these topologies.

Theorern 2.25. The Fricke coorilinales fo: To ---+ R6'-6 is injectiae.

Proof . We need to show that every point fo( lR, t ] ) = (or , " t ,dv, . . . ,a 's- r ,

Co-r,ils-1) in F, determines uniquely the canonical system {oi,0i } of gen-

eiatoriof the normalized F\rchsian model J- for the point [8, E)eTo.

For each j ( j = I ,2 , . . . ,9- l ) ,6 i is obta ined f rom the re lat ion a id i -b ic i : t

with cr' ) 0, and hence oi is determined uniquely by fo(1R,4)' BV the same

argument, 0i U = 1,2,. . ., g - 1) is also determined.

n '(ls'al)U rq, u"paururalap a^eq ellt. 'aro;araq; 'ZZ'Z ureroeqtr slcrperluoc srql ureSy 'cqoqered

sr ,L acuaq pu€ '6 = p + D leql pug e,lr 'I = cq - pD uorlelel aq1 uror; 'snq;'V6'Z e:Uu.Uuielfq O I

6c asnecaq'I = q+ p a^eq e^,r ueq?'I = paa ;r'are11

(erz).on P-I -0" o-L'I-p+c

I-q+D

san€ (2'6) otq (U I'Z) pu€ (II'Z) Jo uorlnlrlsqns

.P-l -o^ -\a-r

,D_I--ocq

.0=tp(I-p)+6qc

'0=ta(r_y-p)*6ec

'0=6?q+6e(t-o)

:ploq suorlenba 3uu*o11o; aql 1eql aurnsse .[eur'.,f1arr1cadser 'p- po* '"- 'q- 'r- ,(q p pue 'a 'g 'o Surceldag

,P+zc _Q)L9tzo

(zrd_

(rrz)

1aB aan '(Ot'Z) pue (9'6) uro.rg'16l pauru.ra?ap e^eq a^,r ecuag '(p -

i/0 - o) - y pue'I

* p'l * e ryql s^rolloJ ll snql 'ZZ'ZvrarceqJ q?rpsrluoc sSlJ 'cqoqered u ,l1eq1 sarldurr qclq/'{'} - (f)zrl ecuaq pue'I = p ueq}'I = D JI'(p - I)y = I - pol€q a^\ (O'Z) p* (9'6) uro.g 'qsrue,r lou saop 6c ro 6o Jo euo lseal le aculs

(orz)(o'e)(s'a)

arrr 'fressacau ;t

'I - ?q - pD 'g) p'e'q'o

= ,L 3ur11nd 'pl

Q'z)

IeS'rldo6po6g = 6noL altsq e/$ 'lld'fall=[U

= lfd'!plr=!6lJ uolleler pluatuepunJ aql urorJ '.raq1rng

'opa6c=6q+6o

eeuaqpue'I 1e lutod pexu e^rlcertle sll mq to'J roJ (ll) uorlrpuoc uorl€zllsruJou aql{g'I < y q}l^{ zy = (z)6! el"q a.&t'J.loJ (r) uorlrpuoc uorl€zrl€rurou eq} /tg'(g'Ul)ot,{q paururralap ers 6d p* to qloq }sq} q ^roqs ol sureuer leq1\

6'aredg e:1crrg 'g'g

50 2. Fricke Space

Notes

For historical and expository accounts of the uniformization theorem, we refer to

Abikoff [2], and Bers [29] and [36]. The original idea of using universal covering

surfaces is due to H. A. Schwarz (cf. Bers [29], pp.264-265). Complete details of

covering surfaces are contained in the books on Riemann surfaces listed in the

notes of Chapter 1.The notion of a F\rchsian group was first introduced by Fuchs in the study of

analytic continuation ofsolutions ofcertain ordinary differential equations ofthe

second order (cf. Ford [A-31], Chapter XI). See also Yoshida [A-113]. For more

details on F\rchsian groups, we refer to Jones and Singerman [A-48], and Lehner

[A-66] and [A-67]. Discrete subgroups of PSL(2,C) are called Kletnian groups,

which are intimbtely related to the theory of Teichmiiller spaces. It is most

regrettable that this interesting subject cannot be covered. Concerning Kleinian

groups, see Beardon [A-11], Berset al. [A-15], Ford [A-31], Krushkal" Apanasov

and Gusevskil [A-61], Lehner [A-66], Magnus [A-70], and Maskit [A-71]. For

relation between Kleinia"n gloups and 3-manifolds, we also refer to Epstein [A-25] and [A-26], Fathi, Laudenbach and Po6naru [A-29], Morgan and Bass [A-76],McMullen [154], and Thurston [231]. Poinca"r6 [A-90] is his collected works on

Fuchsian groups and automorphic functions.For the interaction between ergodic theory and discrete groups, see Nicholls

[A-86], Bowen and Series [47], Morosawa [158], Series [195], and Velling and

Matsuzaki [241].Fricke spaces first appeared in Fricke and Klein [A-33]. For modern treat-

ments, see Abikoff [A-1], Goldman and Magid [A-36], Bers and Gardiner [42],Keen [110], Saito [186], and Weil [243].

For a representation of a Riemann surface' we can use a .9cfioltky group

instead of a F\rchsian group, and we obtain a schottky space instead of a

Teichmiiller space. This topic is discussed in Bers [35], Hejhal [98], and Sato

[188] and [18e].

sa{s4os V - V: t |utddoru ctryl"toutoloy fr"taag (wlo�ru:al s6{crd-z.re,*qcg) '1'g uorlrsodor;

'zz pup- Iz uee/r leq??uDlslp ?rDourod eq1 (zz'rz)d llet e^\.aclr€lslp Jo $uorxe aql seuslles d ryqIu^roqs sr. Il'zz pue rz ?ceuuoc t{clq,$ 7 ut se^rnf, elqeul}leJ II€ sa^ou 3

'arag

,lzl-t rf c 'o'i' i' / -lul = (zz'tz)d

lzplz J - '

las alrlr 'V ) zr tlz slu-tod orr,r.1 fue log'drlauroaS ueaprlcng-uou eJo lapotu e lrnrlsuoc ol clrletu stt{l pesn ?J€culod

'H

.zQl'l - i =

"1'P1tr = "tP

xuleu 9rv?utod eq}sr euo luelrodrur leqloue pue 'rftp * "*p = ,ltpl

-- es'p clrletu ueepqcng aql sl

ureqlJo auo'srrrla{u ..letrnleu,, Iere^es seq {I > lrl I C ) zI = 7 }tslP }Iun eqtr

crrlatr l ?rBcurod 'T'I'8

l(llaruoag rrloqrod/tH pue rlrlatr tr ?rerutod 'I'g

'uolsrnql'A\ fq pasodord dlluacar s€^r qcrq/!\ 'aceds rellnurq)Ial aqt Jo uotlecyrlceduroeelq€1ou e Jo uorlf,nrlsuot eql Jo q)le{s e e,rr3 aaa. 'p uorlcag ut 'fleutg

'urely pup a{?tJd Jo suorle3tlsa,rul Ieclsselc ut ur3tro sl! seq qctq,lr 'sq1Eua1

crsepoe3go sueeru ,(q aceds ueaprlcng ue olur aceds rallntuqclel eql Jo Sutppequaue ssncsrp a,n 'g uorlces uI 'e?€Jrns uu€ureru pesoll e 3o eceds rallnuq?Ial

aql uo 'saleurp.rool ueslerNleq?uad Pallec 'saleurprooc

;o uralsds e auuaP e1'r

6 uorlces ur 'd.rlauroaE crloqradfq 3urs11 'scrsapoe3 Sutu.recuof, esoql fletcadsa'serlredord crseq {pn1s pue elrlau ereculod aql eugep aiu.'1 uorlcag ut'1s.rtg

'{srp }run eql uo crr}eru ersculod aql dq pecnpul fl tlclq^r seceJrns uuetuelguo frleuroa3 cqoqredfq aq1 ;o slcedse auros ssnc$p II€qs a^\ 'reldeqc sql uI

salBurProocuoslarN-Iaqruad puB rt.rlauroa.D rrloq.radfll

t raldBrlc

52 3. Hyperbolic Geometry and Fenchel-Nielsen Coordinates

l f ' . ( : ) l =. +, z € a.I - l f ( r ) l ' = 1 - l " r -

Moroaer, if the equality holds at one point in A, then f is a biholomorphicaulornorphism of A, and the equalitg holds at any point of A.

Proof. Fix a point z in A arbitrarily, and set

w * z.tt\w) =TlZw,

/ \ w - f ( z )l z tw t= , - f z1w '

Then 71 and 72 belong to Aut(A), and ,F(to) = J2 o f " lr(w) is a holomorphic

mapping of 4 into 4. Since .F(0) = g attd

' l - l , l 2F,(o)= ff i f ' (r),we have the assertion by Schwarz'lemma. tr

When we denote by /-(ds2) the pull-back of the Poincard metric ds2 =

aldzl2 /(l - lr l ') ' by /, Proposition 3.1 implies that

f* (ds2) < ds2

and that I.(dt2) - dsz if and only if / belongs ro Aut(A).

Corollary. Eaery holomorphic mapping f : A ------ A satisfies

p(f (zt), f (zzD 1 p(21, z2), 21, 22 € A.

Remark. In general, the Gaussian curaalure /{(h) of a Riemannian metric

h(z)2 ldz lz (h(r )> o) is g iven by

4 fl2logh, h ( h ) = - F d

A simple computation shows that the Gaussian curvature of the Poincar6 metric

is identically equal to -1 on 4.Moreover, we can see that, when a metric h(z)2ldzl2 is invariant under the

action by Aut(A), it is coincideni with the Poincar6 metric, up to a constant

factor.

'y otuo H lo (! + z)/(? - ,) = (z)1, uorleurroJsuertsnlqgl i eqt ,,(q v uo zsp crrleru ?r"curod eql Jo {req11nd aql 1nq 3mq1ou q qclq,tr

, z(lu'l) = ,rrp"l'Pl

3ur11as,(q paugap sr g aueld-geq .raddn aql uo {sp crrlaur ar"curod eq&'tlrout?[

''V Jo rr"qns e q LV uo slurod oall i(ue Eurlcauuoc crsapoe3 fre,ra'6'9

uorlrsodor4 ,tq 'teql a?ou eJaH ',0 fq uorlce eql rapun luerrslur sr f,y uxe aq;'L p'V sDr€ eql palpc sr Ve oI 1euo3oq1.ro sr pue slurod asaql q3no.rq1 sassedqcrqa,ll luaur3as auq eql ro alcrr? eql Jo y u1 lred aql l"ql Ip?eU

'Vg uo Lo prte L.r,

slurod paxg Irurlsrp otrl seq l, 'crloqredfq 4 (VhnV 3 ,L uaqal '1eq1

lpcag

tr 'lzz'o)-

7 luaur3es euq eq? qlr^r lueprcurof, sr Cyr fluo pueJI (rh = (zz'1)d acuag

.2, - Ixp7,

eleq era 'zz pue g Surlcauuoo C trts pesolc frala ro;'r".II'U f d elq€1lns qYal. (V)7nY P

zlz -t

#eP = Q)L

luauela ue fq tuaql Sururrogsuert /tq '0 1 zz pue 0 = rz 1eq1 etunsse deuren'(V)1ny fq uorlce ar{l repun lu€rrelur sr crrleru ar€curod eq1 acurg /oo.l4,

'v loVg fi.topunoq eql oI 1ouo0or17.to sl puD zz puo rz q0norqt sassod t1cn1m 7uau,6aseuq eql ro el?Jr? eyy to ctoqns o s, puD anbtun st 7r |teaoanory 'V ul zz puo rz|utTcauuoc crsapoa0 o slsNae ere1l 'V ) zz'rz fi^to4tq.to ro,I 'Z'g uorlrsodor6

'(C)l = (zz'rr)d

eleq e^r JI '9z ul zz pve Iz $ullcauuoc (cr.r1eru

gr€oulod aq1 o1 lcadsar q1ra,r) ctsapoe0 e 'V ul zz pue rz Surlcauuot'g cre pasolcelq€Urlcar € IIef, e \'V ) zr 'rz s?ulod orrr1 fue rod'(r)/ fq 1r elouap pu€'Cp y76ua1 cqoqtadfr,tl aql sp "[

lV, ell.'V ur , ]re pasolc alqegrlcer fre,ra rog

scrsaPoaD 'z'T't

,{r1auroag cuoqradifll pu" f,rrlel{ gr"f,u-Iod 'I't

",ol ,W"[

t9

54 3. Hyperbolic Geometry and Fenchel-Nielsen Coordinates

3.L.3. Hyperbolic Metric on a Riemann Surface

Let it be a Riemann surface whose universal covering surface is biholomorphi-cally equivalent to 4. Consider a Fhchsian model I of ,R acting on /. Letr: A - r? be the projection of 4 onto R = A/f. Since the Poincar6 metricds2 is invariant under the action by .l-, we obtain a Riemannian metric ds2p onR which satisfies

r- (dszp) = d,s2 .

We call this dsft the Poincar6 metric, or the hyperbolic metric on R.Now, every 1 e f corresponds to an element [Cr] of the fundamental group

r{R,po) of ,R (Theorem 2.5). In particular, 7 determines the free homotopyclass of C7, where C,, is a representative of the class [Cr]. We say that 7 coaersthe closed curue Cr.

When j € f is hyperbolic, it is seen that the closed curve -t, - A-, I 1l ),the image on .R of the axis A, by zr, is the unique geodesic (with respect to thehyperbolic metric on ,R ) belonging to the free homotopy class of Ct. We call L-,the closed geodesic corresponding to 7, or to C.,.

Proposition 3.3. Lel R be a Riemann surface with uniaersal couering surfaceH , and 11 be a Fuchsian model of R acting on H. Let

a z * btk) = a , b , c , d e P - , a d - b c = 7 ,

c z * d '

be a hyperbolic elemenl of 11, and L, be the closed geodesic on R correspondingto 7. Then the hyperbolic lenglh I(Lr) of L, satisfies

t. '(r) - @+ d)2 = 4cosh2 e)

Prool. Since t(L-r) and tr2(7) are invariant under the conjugation of 7 by anelement ol Aut(H), we may assume that 7(z)

- )z (.\ > 1). We may alsoassume that o - t5, b = c =0, and d = I/\5.In this case, we have

= log ) = 2 log a.

Hence we have the assertion.

3.1.4, Pants

Consider cutting a Riemann surface r? which admits the hyperbolic metric by afamily of mutually disjoint simple closed geodesics on R. Let P be a relativelycompact connected component of the resulting union of subsurfaces. If P containsno more simple closed geodesic of .r?, then P should be triply connected, i.e.,homeomorphic to a planar region, say

((L.t) = Ir^ +!

'd louoNeuelse ueslerN aql Pellet sr d Pu€'d Jolaweq ueslerN erll Pall€l

st 2r 'f11en1rq€H '-dJ fq paurunalap flanbiun ,, j '.pto^ reqto uI 'd

Jo slued

;o rred anbrun eql s! plq^{ '2' 3o aee;rnsqns € s'e pereplsuoc sl d

',,(pee13

'(t'g '31.{ eas) ; Jo lapour uelsqcqil e q d.7 pue 'pelcauuoc ,{1dr.r1ure3e sr 2'

'acue11 'lueuodtu6r ,t.repunoq qcee Suop uorSar palceuuof, .{1qnop

elq€lrns e Surqcelp fq 4'uro.r; paul€tqo areJrns € sI dr uaql 'a,t/V - d las

ql/d - d pue'suotleurrogsuerl

crloq.rad,tq om1 fq pele.reue3 dno.r3 aa.g e s1 d.7 uaql 'd = ({)t }€tI} q?ns J Jo L

sluetuele 1e yo Surlsrsuoc J Jo dnor3qns eqt dJ fq alouaq '(d) r -! Jo lueuoduroc

pelceuuoc e aQ d 1e1'uorlcelord aq1 aq JIV = U - V i )L pve'7 uo 3ur1ce

U Jo lapour uersqcqE e aq J 1e1 'fpre.r1rqr" g 3o 7 slued ;o .rted e xtg

'r'8'tIJ

'U uo rlsapoe3 pasolc elduns € sl 2I ul d Jo frepunoq

e^rleler eqt Jo luauoduroc palrauuoc frarra 3r Pue Palceuuoc i(1dt.r1 q d JI g'

1o sTuod;o .rted e U Jo d eD€Jrnsqns lcedtuoc f1errr1e1e.r e IIef, e^\ '.re1;eara11 'g

Surppnqa.r .ro; secerd lseilerus eql Jo auo s€ pereplsuoc aq ue? d et"Jrnsqns e qcns

-{z>lzl}=04 ({i t rr - ",}^ {i > rr +,r})

4J/V:d

ccfrlauroa.g rqoqraddll Pu€ f,rrlel{ grsf,urod 'I't

3. Hyperbolic Geometry and Fenchel-Nielsen Coordinates

3.1.5. Existence and Uniqueness of P'-ts

We shall discuss the relationship between the complex structure of a triply con-

nected domain J? and the hyperbolic structure of P, the unique pair of pants of

O, induced by the hyperbolic metric on O.Let L1,L2, and..L3 be the boundary components, which a.re simple closed

geodesics, of the pair of pants P. Let J-e be a Fuchsian model of the domain

O acting on A. Then'i-s is a free group generated by two hyperbolic trans-

formations, say, 7r and. 72. We may assume that 71 and 72 cover .L1 and L2,

respectively.

Theorem 3.4. For an arbitrarily giaen triple (ayaz,as) of positiae numbers,

lhere erists a triply connected planar Riemann surfoce Q such that

t ( L i ) = a 1 , i - 1 , 2 , 3 .

Proof. We prove it by constructing O explicitly.Let Cr be the part of the imaginary axis in A. Fix another geodesic, say C2,

on 4 such that the Poincar6 distance between C1 and C2 is equal to a1f 2. On the

other hand, geodesics on 4 from which the Poincar6 distance to C1 are equal

to oBf2 form a real one-parameter family (i.e., the family of circular arcs Cl

tangent to the broken circular arc in Fig. 3.2-i)). Hence there exists a geodesic,

say Cs, in this family such that the Poincard distance between Cz and Cg is

equal to a2f2.

Fig.3.2.

Next, let 21 arrd z2 be the points in 4 uniquely determined by the condition

p(rr, r) = ?,

zr e Ct, zz e Cz.

Let L\ be the geodesic connecting 21 and 22. Similarly, let {z3,za} and {rs,re]1be the pairs of points uniquely determined by the conditions

lsrll eutrrnssp feur e,r. 'r(.ressacau.l uorleEntuot-(g)7ny ue 3ur:1et 'asodrnd srqlrog'r={{{o},(q peururralap,{lanbrun are zL pue I,L }eql ^\oqs o} seclsns U'e? sraloc ,_(tf o zL) - eL teql pue (e 'I = 1) 17 s.rairoc {1, ?sqt arunsss feur aa,r'ara11 '0.7

Jo srolereuaS;o rualsfs e aq {zt 'I.L}

leT 'g eueld-y1eq raddn eq} uo

3ur1ce d Jo lepour uersqr\{ e eq 0J pu€ 'd Jo uorsuelxe ueslarN eql eq d ?c,-I'd

Jo (g'Z'l = f) f7 lueuoduroc frepunoq aq1 ;o q1Eua1 crloqlad,(q eql eq lp p"I'fy.rer1rq.re uaarE sr 2, sluedgo ned s teql esoddng ('IIt'[Ott] uaay'93) 'too.r4

'4 lo sTuauoduoe fi".topunoq perepro aq1 lo st176ua7 cqoqtedfrq aqy fiq

p?aunrepp fryanbrun sa 4 syuod to .ttorl o to ernlrtuls aelilutoc ?ttJ .g.g ruoJoaql

tr '(g'g '81.{ eas) ace;rns peilsap " sr {Ji snqtr '0J ,tq uorlce eql rapun (O)tbnOgo ,trepunoq eq1 Surf;rluapl fq paurclqo ?as eql Jo rorrelur eql sl (J ;o 2, sluedyo rpd anbrun arl? lstll pue 'pelceuuoc f1du1 sr. oJ /V = U leq+ realc s-r lt

'8'8'ttJ

.z,L pue rl, aseqt fqpele.rauaE dno.rE aq1 aq o.ir 1e1

'(V)l"V t" sluauele arloqrad,tu are zl, pue rl, ueq,L

'Vtoeb-zL tebsV)-rl-

les 'eslrlurod fg Eur,rraserd 3 go ursrqdrour

-o1ne crqd,rouroloq-rlu" aqt ''e'l'f^2 o1 laadsar qlr^r uorl?auer eql;q (g'Z't = f)lh p1 't=[{!,1'!c} fq pepunoq uo3exaq cqoqrad,(q pesolt eq? eq o p"r

('(fa'g 'ft9 eag) 'ez Pue ez Pue'?z

pue 8z Eurlceuuoc scrsapoaS eq1 'f1a,rr1cadse.r '!7 pve 1I ,{q alouaq 'flelrlcedser

'rC)sz'eC)sz

'88888C)vz(zC)Ez

= (sz(s2)i

- (vz'es)6

,7,tp

,zZ9

L9i(rlauroag ruoqradi(g pu" rrrlel l ?r"3ulod 'I't

58 3. Hyperbolic Geometry and Penchel-Nielsen Coordinates

I t ( z ) = \ 2 2 , 0 < ) < 1 ,

/ \ a z * b.r2(z) =;i i , , ad - bc - I ,c ) 0,

and that 1 is the attractive fixed point of 12, or equivalently,

a l b = s a i , O < - ! < t .c

Then we see that fz(m) = a/c ) 0 and a + d > 0, since the middle-point(a - d)/(2c) of two fixed points of 72 has a value less than Zz(oo).

Next, write

, \ - 1 , , d z * b(73 ) - ' ( z ) = -uuA , ad -bz - 1 -

Since (73)-l = j2o 7r, we may assume that

6 , = a \ , 6 = b 1 \ , E - c \ , d = a 1 > , .

In particular, d > 0. Moreover, the middle-point (6 - a11pe1of the fixed pointsof (73)-1 has a value greater than (73)-r( x) = d/8. Hence, a + d < O.

On the other hand, by Proposition 3.3, we have

() + l/r)'� =4 cooh2 (+) ,

(a + d)2 =4 cosh2 f +) ,\ 2 / '

@ + J:2 =4 cosh2 (+)\ 2 /

Therefore,'y1 and j2 are uniquely determined by {or, az,as}. tr

We have proved that, for any triple of positive numbers, there exists a pair ofpants admitting a reflection (induced, for example, by rlr) such that the hyper-bolic lengths of the ordered boundary components are the given triple (Theorem3.4), and that it is uniquely determined by the given triple (Theorem 3.5). Thuswe have the following corollary (see also Fig. 3.4).

Corollary. Eaerg pair of pants P has an anti-holornorphic automorphism Jp oforder two.

Moreoaer,Lhe set Frr= {z e P lJp(r )= z} of a l l f ixedpoin ls of Jp consistsof thrce geodesics {Di}|=, in P satisfying the following condition:

For euery j ( j = L,2,3), Di has the endpoints on, and is orThogonal to, bothL i and L1 ; r , whe re L+= L t .

We call "Ip described in this corollary the rc,fl,ection of P.

.6J

uo selDuzpron ueslely-leq?ury pelle;. sI sel€ulprooc;o rua1s.{s € qcns 'sacerd aqlan13 o1 pasn sralatuered 3ut1stall pellec-os arll Jo las aql pue slued olur uorysod-ruocep e^oqe eql ur pesn scrsapoe3 11e;o sqlSualJo les aq1 ;o.ued aq1 'rg acedsrellnurqcral aql roJ sel€urproo?;o ue1s.{s e se '.reptsuoc ue, a^\ 'ecue11 'f1qe1tns

secard 3ur11nsar IIe 3urn13 fq palcnrlsuocer sI U leql r€elc sI lI 'g'g ura.roaq; ,{q

1r Jo scrsepoe3 ,{.repunoq yo sqlEual cqoqradfq eql Jo a1dtr1 aq1 fq paunurelep.{lanbrun sl g. Jo slued ;o rred qcea Jo ernlf,nrls xelduroc eq} }eql II€reU

'p. yo qued 3o .rred e eq plnoqs ecard drarra ueql '1as uado Surureurer eql ulpaureluoc U Jo s)rsepoa3 pasolc alduns eroru ou are areql uaq1yg, uo {sp f,Ir}aurcrloqrad.rtq aq1 o1 laadser qllr* srtsepoa3 pesolc aldurts 1u1ofs1p flenlnur 3uo1e y3ur11nc raprsuoc'eroyaq sy'(e {) f snua3Jo af,eJrns uu"ruerg pesol? e aq Ur lerl

uorlrsodtuocaq slusd'1''Z'e,

'.re1deqc $r{} Jo rapulqurer aq1 ut fleeq suollJasse eql esn

[eqs e,lr q3noql 'g .ra1deq3 lr]un rueroer.{l sqtJo;oord e Sur,rr3 auodlsod a6

'g-fgll o7 ctr1d.r,ou.to?uo?! puD e-oell a!uzD'tuop D s! 6l acods aqu.r,tr eqJ (g1'g uraroaqa) 'uraroaql s.rallnurqtraJ

'.{1e,rr1rn1ur reql€r 1nq 'o3e aurrl 3uo1 e pelrecuof, sehr uaJoaql Euro,o11o;

eq1 're,roarohtr'e-6etlJo lasqnse sl tdr eceds e4crrg aql'flsnoue.rd pa1e1s sy'.{rlaruoa3 crloqred,tq Sursn ,,lq 6g o1 saleurproo) Jo

ad.{1 raqloue acnpor}ur a,r. 'uotlces slt{l uI 'saceJrns Jo slepotu uelsqcnd Sursn ,{q

'eceds ueeprpng l€uorsueurp-(S-0g) learJo (eceds e{clq aq1 parueu) tg lasqns* * (Z ?) f snua3 ;o

tg aceds rellnuqtlal aql peluasardar am, '6 reldeqS u1

sa+BurProoc uaslarN{aqruaJ'z'8

'?'8'ttJ

sat"urProoc uaslarNlaqruad'z'8

60 3. Hyperbolic Geometry and Fenchel-Nielsen Coordinates

Now,. we grve a precise definition of these coordinates, and verify that theygive a system of global coordinates on To.

For this purpose, first fix a point [.R,.D] of ?r. A set 4 of mutually disjointsimple closed geodesics on .R is termed matirnal if there is no set 4' whichincludes 4 properly. We call a maximal set .C = {fi}j!, of mutually disjointsimple closed geodesics on rt a syslem of decomposing curaes, and the familyP = {PxW' consisting of all connected components of R - UltLi the pantsdecomposition of R corresponding to L.

Emmple. When g = 2, there are the two kinds of pants decompositions shownin Fig. 3.5.

Fig.3.5.

Proposition 3.6. Let L = {li}l=, be an arbitrary system of decomposing

curaes on a closed Riemann surface R of genas C (>-2), and letP = lPxlf=tbe lhe pants decomposition of R concsponding to t,. Then M and N satisfy

N = 3 g - 3 a n d M = 2 g - 2 .

Proof. Cut r? along an element Ly of L. Let n1 be the number of connectedcomponents of .R - .t1, and 91 be the sum of genera of all connected componentsof R - tr1. Then we have

9 r - n r = ( g - 1 ) - 1 .

Clearly, the number of boundary components of ,R - -t1 is two.Moreover, we can see inductively that, whenever we add a cut along a new

element of 4, the number of boundary components increases by two, and thesum of genera of all connected components minus the number of connectedcomponents decrea^ses by one. Hence, we have

g M _ _ 2 N a n d 0 _ M = ( g _ 1 ) _ N ,

.l.r!t! "^ = (7)!6 (t)!t-" -

1ag'(1ou rc (7)17 Jo 1eql qlrar alqrleduroc s1 (3)fuJo uorteluerro eql raqleqa olEurproace a.,rrleEau ro a,rrlrsod sg (3).r.r. 1eq1 os) (l)l,l,l" (l)f, qt3n"t cqoqred,tqpau3rs eql eusap ust e^l 'lI

lo l"ql uro+ peururalep uorleluarro l€rnl"u eqlseq (l)fZ acurg'(3)z'fc o1 (3)t'fa urory (3)f7 uo f,tre patuerro eqt aq (t)fap1

.(ic'ta Jo uorlcagar

aqlgo lurod pexss sl (U 'I = t) (1)t'!c qcee uaql'(l)f7 uo Ot't1;o lurod aq1(ic'ft fq pue '{t3ua1 prurunu qtyn (1)r'12, ur (7)t'!7 pue (1)f7 Eururof crsapoa3aq1 (1)r'fc, i(q alouaq 't'17 ol Eurpuodsarro) (i't'!d;o luauodruoc frepunoq eq1eq (7)t'!7 p"l'1'14 u1 tr'!7 fes 'luauoduroc f.repunoq raqloue pue !7 Sururofr'f6. crsepoe3 aq1 yo f7 uo lurod pua a{} sl (Z,I - q) t'fc qcee }€ql IIeceU'f1a,rr1cadsa.r 'z'!4 pue r'14 o1 Surpuodsa.uo? (tU;o slued a.re qcrq,u,) (l)fZt=Jn- ? Jo sluauoduroc palcauuo? aql aq (iz'fa pue (1)t'f4' 1a1

'f, pue l fra,re rog''d.rl I {rara.rog 1o1 Eurpuodserroc ';;o lurod aql aq l,3'rA) 1a1 'a.ro;aq sy

'(g'g'3t.{ aas) f7 uo uol}sluerro us osls "g'r'fa,tq t1alouep pu" '(e'I = {) r't, qcea rc1 !7 uo Y Jo lurod paxg " e{"I 'g'g ureroaqJo1 fre11oro3 eql ^q 'flarrrlcadse.r 'zf pue r/ uorlcager aql lrurpe z'!4 pue t'14

leql II€lsU 'z'f4 - r'.r2' a.raq,r,r es€l aql ,&roll" e^r areH'lueuoduroc drepunoq e

g" f7 Eul\eq d ul slued;o s.rred oiu.1 aq z'!4 pue r'fd 1a1 '/ fra,re roy 'lxaN

sralaruerBd Eullqar;'g'Z'g

'6d uo cr1fi1ouo1oa.r, st (l)h uo4cunt y76ua7 nsapoa| fi.taag 'Z'g BuruxaT

serTdur 6'9 uorlrsodord ueqy t7 rc1uot1cun! vtf uq ?*?po;fflttftTop�H iUuo 'flluelerrrnbe.ro) 6g uo uorlcunJ € se (1)f7 raprsuoc eM'(ib fq fldrurs Ol7 p((l)!l)l qfual cqoqrad{q eq} elouep eiu'f l(ra,re pue td ur 3 itre,re ro;'aro11

'f, fra,ra q (!1)tg sreloc qcq/rt ,J Jo luatuala u€ Jo slx€ eq1 ;o uotlcafo,rdaq1 sr (1)f7 leq? eloN'1fq paluesardar rgrJo lepow u"rsqcr\{ eqt aq tJ ta1

.rar uosalrnc Sursodruocap;o uralsfs e sr t--;f{11;t1\ = rj'eoue11 ',1

+ | uaqar lurofsrp.r(11en1nur ere (7),17 pue (3)f7 l€rl? pue 'eldurrs sr (1)f7 ?€rll aorls ot llnclgrp touq lI

'? uo (!7)r1 a^rne pasole arlt Jo ss€p fdolouroq eerJ aql ur crsapoe3 pesolcenbrun aql eq (l)fZ f"t 'J ul !7 {ra,re rog '(y'1 uraroaq; 'Jc) tU - A : tt

ursrqdrouroauroq Eur,rrasard-Eur4retu e alel tr(1arue11 'tgr uo selrnf, Sutsodurooap

Jo I=J{(t) l?} = ,J urelsfs e ,tlanbrun aunurelap uec er$ 't".II '? o1 Eurpuodserroct; ur lurod aqt lt3' "U] ,tq alouep e$'6,tr a*ds aqcrq4 eqt ul t fra,ra rog '3. uoselrnc Eursodurocep f" tlf{fZ} -

7 ure1s.{s e pue '6J lo [g'g,] lurod e xld

suorlcur\{ q1Eua1 crsapoaD 'Z'Z't

'uorlrasse aql ,(1dun qcqaa

seleuProoc uaslarN-Iaqf, u3,{'z't I9

62 3. Hyperbolic Geometry and Fenchel-Nielsen Coordinates

F ig .3 .6 .

Then 0i(f) is well-defined modulo hr. We caII01(t) the twisting parameler withrespect to L1.

Lemma 3.8. For euery j, exp(id1(t)) is well-defined and real-analylic on Fn.

Proof. Ftx 1. For every f in .Fr, let fi be the Fuchsian group represented by t.Take an element of 4 which covers tr;(l), and denote it by 71 (t). Next, for each&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&(= 1,2),Iet 7i,x(t) be the element of [ which covers L1,x(t) and satisfes that

the geodesic Di,n(t), connecting ,41(f) and A1,*Q) with the minimal length, isprojected onto Di,*(t), where,4i(t) and A1,r(t) are the axes of 7i(l) and T,x(t),respectively (see Fig. 3.7).

Here, we may assume that the fixed points of 7i(t), t i ,t(t), and 7i,2(t) movereal-analytically on d. Hence, when we take aconjugation of ,Q by an elementof Aut(H) so that 71( t ) goes to fu( t ) (z) =. \ j ( r ) .z( \ ( t ) ) 1) , the f ixed pointsof fu,*(t) corresponding to 7i,1(t) move also real-analytically on Fo for each #.

Now, c1,1(t) is the projection of the end point Zi,r(t) "t fu,x(t) to A1Q).Ifence, if we show that 6i,x(t) moves real-analytically on Fr, the assertion followsby the definit ion of 11(t) and Lemma 3.7.

To show this, fix /c, and let p1 and p2 be the fixed points of ii,i(l). Setc13(t) = iv* (v* > 0). Since

'7* (ry)' = (o'to')' ,we see real-analyticity of ci,r(l).

Dt,z

D;,r

(P. ; ,1: Pi ,z)

.([tqz] 1.rad1o44 pue ,h-yl

.Uo{lqy ';c) ursrqd.rouroe$lp e sl g_ogtl x ,_rg(ag) - 6l : 4i

'dgenlcy 'IrDur?A

'(sa,r.rnc Eursodurocep;o 7 uralsfs eql qtyr\ 'to) d uotltsodtuocap slued aql

rl?-rrA paler)osse t;;o selouxproox ueslery-Ieq)uef ^seler;rprooc asaql II€r elA

.6a uonuaq puD'6,I uo sa?Durproo? 7oqo16 lo ue7sfrr o saat6 4i'.to7nc4.r,od uI 'e-oellX

e-re(+U) oTuo 6g to rusttld;oruoeuoq e s? 4 6utddnu, s?ttJ 'OI'8 tuaroaql

'uaroaql 3uralo1o; eq1 erro.rd ,r.ou ilBqs aru 'le,roelotr41

'uf uo ctTfipuo-pat st

((l)"-u"a' . . ., (t)r 0, (t)e- ueT . . ., (t)rt) = (t) rt

ueqJ '! tuaaa ut 6g uo (7)lgnTatuo.tod 6ut7stm7 aqy to qcun"tq snonurluo? panlna-e16uts D ottr '6'g eurtrrarl

:3ur,r.lo11o; eql e^€r{ e,lr snqtr '(uraroaql ftuo.rpouour eql) t/ uo r{ou€rqsnonurluoc panp,r,-e13urs € s€rl (3){6 frarra',{13urp.roccy 'uleurop paltauuof, .{ldurrse s-r tJ l"q1 sale?s (91'g ura.roaql) ura.roaql s(rellnuqclatr (pueq raqlo aqt uO

'{f = Itl I C > t} = rS pue tO < t lu> t} - +lI 13s eA\ereg

' (((r)t-rtOl)dxa' . . .' ((1) t6r)dxa' (t)e- aq . .'' (t)rt) = (t)'t

:e-re(rS) x e-oe(+tI) - 6tr : 4i Surddeu cr1.,{1eue-1ear e Paugep eleq ax\ '.re; o5

salBurPlooc uaslalN-IaqcuoJ'v'z' I

't'8'ttd

Q|'r,

(t1z''t (t)z''g

U7t'rt

t9sal"urprooc uaslarN-leqf, uaJ'z't

64 3. Ilyperbolic Geometry and Fenchel-Nielsen Coordinates

Prool. First, we show thatitrr is injective. Suppose that fr(tr) =rir(t) for somet1 and t2 in Fo. Let Ra, be the Riemann surface represented by t; (with thenatural marking), and A = {P}(i)};s=;" b" th" pa^nts decomposition of &,correspondingtoP, for each f (= 1,2).BV Theorem 3.5 and the assumption,there is a conformal mapping, say g&, of P7,(1) onto P1(2) which respects theboundary correspondence for every /c. Moreover, the proof of rheorem B.b impliesthat

dsl = (si4.@sl).Here, dsl (f = 1,2) is the hyperbolic metric on &,.In pa.rticular, every 9r is ahyperbolic isometry of the closure of Pi(l) onto the closure of Pp(2). Since

| j ( t r ) = ? i ( t r ) , j = 1 , . . . , 3 9 - 3 ,

all g1 can be glued together into a ma^rking-preserving homeomorphism, say h,of rt1, onto -R1r. Since lr is holomorphic on r?1, except for a finite number ofanalytic curves, so is h on the entire .R1, by Painlev6's theorem. Hence, h is abiholo-morphic mapping of -R1,, which implies that t1 = lz. Thus we have provedthat P is injective.

Next, we show that f is surjective. For this purpose, we begin with fixing apoint (4a1, .. . ,eas-s,e1t. . . ,ass-z) of (R+;sc-a x R3s-3 arbitra.ri ly. For everyP3 in the pants decomposition P of R, denote by {Il,i}i=, (C .C) the boundarycomponents of Pp. From $1.5, there is a unique pair of pants, say Pto, such thatthe triple of the hyperbolic lengths of the boundary components of Pf is equalto the given triple {ou,i}i=r.SetPt= {P;}ir=lt.As before, let pil and pi,2 bethe elements of P neighboring each other along .Li for every j, and let pj,xbethe element of P/ corresponding to Pip (l = I,2). Let Ci,2be the point oii th"boundary of Pjp correspondingto ci,t for every j and l.-

Now, by gluing Pj,1 and P/,2 suitablV along curves corresponding to ,Li forevery j, we obtain a Riemann surface, say r?'. We need to choose a suitablegluing (and a-suitable marking of .R') so that R/ corresponds to a point t, ofF , such tha t f ( t / ) i s equa l t o t he g i ven (a1 , . . . , a1s_B ,d l , . . . , ass_a ) . Th i s canbe achieved by gluing Pr{,1 and Pj,2so that the twisting parameter becomes thegiven ai for every j.

We shall explain this procedure more rigorously by using Fuchsian models.In the rest of this proof, we consider only the case where Pil t' pi,z, for theother case can be considered similarly.

Fix j, and let 4,r be a Fuchsian model of the Nielsen extension P|,r "t fj,tfor each &. Here we assume that every 4.,r acts on the upper half-ptanl f , airdthat the transformation

l ( z ) = \ 2 , ) - e x p a i ) L

belongs to both 4,r and li,z, and 7 covers the boundary component, say Ll1r, ofPj,r corresponding to.Li for each & (& = 1,2). We also assume that the niluralorientation of the axis ,4 = {z € Hl, - 'iu,y ) 0} of 7 corresponds to theprescribed orientation of .ti, and that the point i € /l lies over cl,x with respectto li,* for each /c.

'Z-6?,'...'I = { "la =(qa)rl

l€tll q?ns ,U oluo A p rt tusqd.rouroauoq e xrJ 'eroJeq * "'fa pue r'{" 3urn13fq paurclqo ec"Jrns uueuerg eql sq ,A 1eI"tr;o dlrrrrlcaf.rns;oyoord er{l Jod

('r 'dtq3 '[tz-f] tl{se4 acu€tsul roJ eas'sruaroaqluorl€urquo) .rog '(saaeg.rns om1 Surlaauuor .rog) uuoeql uorlDurquoc s(uNelN seulrou{ flpcrsselc q slqtr) '!/A

Jo uorsuelxe ueslarN eql Jo Iapou u€rsqcr\{ e sr

,-gz'!J9 pue I'lJr {q paleraueE dnorS uersqf,nd aq1 'sprorn raqlo uI'{ rl"€a roJ,l'! ,

tt'!4 uo '?"p ,It!^ lueprcuror s (a,una f.repunoq Eurureura.r {.raaa Euole'}'I$ ulr'la p uorsuelxe uaslarN aqlJo uorlrnrlsuor eql ur pasn se^r qf,rq^\'ureurop Eurrelq"lrns e 3urqce11e fq lnt urorJ peurelqo ec"Jrns uusuarg eql ''e'l) fr14 ureurop3ur11nsar ?ql Jo uorsuelxe ueslerN eql uo f,rrleru cqoqred,(q "ql'(H)?nV 3 9 acurg'fo olnpou fp3o1 ot lenba sr z'!p oI r'fa uorJ {fua1 uor}elsusrl arlt }erll qcns

1z'!,7 pue t'l,l to uorlecgrluapl eql ,(q) z'!4 pue t'fa p 3urn13 e e^eq e^{ ueqJ

'8'8'EtJ

('g'g'q.{ aag) 't'1Jo tuetuele rrc se pareprsuoo,LJo y srxe aq} uo (")g qft^

z'!tr p lueutuele ue se pereprsuoc ,LJo y srx€ eql uo z f.tete,tg11uap1 '(n)WV lo

0 < lp 'ztp - (z)g

luauale eql raplsuoo pue'(r,6f lofo)dxa = fp 1eg

|;e 'nJ

99sal"urProoc uaslarN-Iaqf,uad'z't

66 3. Hyperbolic Geometry and Fenchel-Nielsen Coordinates

Then [rR',h-(t)] determines a point of To.Let l 'be the corresponding point of.Fo. From the preceding construction, it is clear that

and that

(Note that, at preseni, we cannot say that 01(t') = ai, because the choice ofbranch of 0i is not unique.)

Hence, letting Ti; R' ------ R'be the Dehnlwist with respect to trl , the curveon.R'corresponding to.ti, we can find integers ftrt. ' . ,f l3s-3 such that

[R:, (Ti 'o . . .o r t i : i " o h).( t ) ]

corresponds to a point, say ttt, which satisfies

, i r 1 t , , 1 = ( o r , . . . , a s s _ s , o t r . . . , o s c _ s ) .

Thus we have shown that fr is surjective.Here, the Dehn twist Q with respect to.Lj is, by definition, a homeomorphism

of R' onto ,? corresponding to the following surgery: cut R' along L! a.nd reglueafter a rotation of 2zr (see Fig. 3.9). Note that applying Ti, we can make thevalue of di increase by 2r while every other 0i U'+j) remains unchanged.

Fig.3.9.

Now, we have proved that fr : Fo -------+ (R+)sr-a x R3r-3 is bijective. ByLemma 3.9, tit is also continuous. On the other hand, Teichmiiller's theorem(Theorem 5.15) states that Fo is homeomorphic to R6c-0. Hence the followingtheorem, Brouwer's theorem on invariance of domains, implies thatV is actually

l i ( t ' ) = a t , i = I , " ' , 3 9 - 3 ,

) . t

01( t '1 : ^ : I "e d1p o1) (mod 2r ) , j = t , . . ' ,39 - 3 .uj

n

o o0

twist

a homeomorphism.

aldurrs (anbrun) aql aq lV m 'raqtrr\{ 'IV tq tr alouap pue '17 s])asre]ur qcrq^r

fr14 ur crsapoaE pasolc aldurs e xrg 'z'f4, n lI nt'!4 - lr14 ps'f fre,ra rog

('7"{d = I'fd teqf as"r eq} ̂{olls e,lr'ure3y) 'luauoduroc,trepunoq e se lI Eur,teq

dJo slueuele eq1 a'f4, pue I'ld' fq elouep 'J > lI {rarra rog '7 o1 Surpuodserroc

U Jo uorlrsodurocap slued aql aq d +aI'U uo selrnc Sursodurocap go 7 uelsdse pue (6 l) f' snua3 Jo Ur ersJrns uu€ualu e xg 'uorlces snor,rerd eql q sY

'uorl€zrJlerue.red;o pur{ raqloue arrrS leqs aira. '21'9 uorlrs-odo.r4 ur 'ra1e1 'suorlrunJ

{lEua1 crsapoa3 o^\} qll^r sa}eurproof, ueslerN-leqruegpaxg q relaurered 3ur1sral1 qcea Surcelde.r fq '6g go slurod aleredas suoll?unJ

{}8ua1 esor{^\ scrsapoa3 pesolc aldurrs 6 - d6 Jo }es e lcn.rlsuoc aa,r 'a.re11'(FOt] IIBAToS pue eleddag ';c) elqrssodurr

q slql '.rala,no11 'oJuo selsurprooc pqo13 a,rrE suorlcuny qfual asoql( srrsapoa3posolt aldurs g - 69 Jo les € 6?ilr ererll ;t elqsrrsap lsoru eq plnoa lI

'areJrns eqleururalep sqfual crloqradfq asoq^\ k ?) 0 snuaS;o af,eJrns uuetuarg pasol, e uoscrsapoe3 pesop eldurrs;o 1es e Surpug;o uralqo.rd eql replsuor e,n 'uotlaas qql uI

turppeqtug ulalx-a{rl44'g'g

')

o1 lcadsar {1a r_4;o flmurluor;o;oo.rd }cerrp € arlr3 osle lleqs a1ys3urddeu

IeuroJuocrsenb Sursn fq g reldeq3 ur adfl $rll Jo uorleruroJep eq1 ele3rlsaaut

Upqs a1yuotTotu.totap u?slerN-Ieycuadr e pelpr sr elrnf, " qf,ns ,tq pelueselderac€Jrns uu"tuerg e Jo uorlerJel eql 'tJ ur eAJnf, snonulluot e o1 spuodsauoc

({U > I | (€-6sot ...rr*ln,J't-!n, ...,Ir2,8-6tD, ...,Ip)}) r_4

e3eurre.rd eql '0I'g ueroeqtr dg 'sralaure.red 3ur1sr,ra1 eqt yo fg

euo e{€l pue'n-rs?I xe-ne(+u);o (e-Eeo' ...'rD's-68D ' . '. '1o) lurode xrg

! 'uollJasse aql a^eq arrr "{rerlrq.re

sr f erurg'IO uo snonurluoc sl r-d pue'O;o lurod rorJelur ue sr f '.re1nct1.red

uI'IO qlr^r luaprf,uroc eq plnoqs d dq g Jo g lul rorrelur eqf Jo (g' 1u1)d e3euraq1 'acue11 'pelrauuoc s\ fI -,lI ler{} u^roqs eq u"? ll

'pueq retllo eq} uO'/s'

Jo uleluop rolrelxeeql pue ureruop rorJalur eql 'fla^llredsar 'zO pue IO {q alouaq 'slueuoduroc

pelcauuoc o,r.l s€q ,S - .rll (uraloeql s(u€pJof leuorsueurp-, "ql fq 'acue11

'rll ul 'fle.rrrlcedser 'ereqds pcrSolodol epu€ IIeq pesol) 1ecr3o1odo1 e erc (gg)dt -

rS pue,g're1ncr1redu1 '(g)dt = ,goluo €r go ursrqdrouroeuoq € s.r g uo o1 yo uorlculsar aq1 'lceduot sr Br aculs'r reluar qll/rl 5' ileq pesop e f1tre.r1tq.re xg'(f)r-d - o las pue'f1t.re.r1tqre

O q n lurod e xIJ ('[98-y] ueu^\eN pue '[ZI-y] sreg 'y3) '(qc1aqs y) too.r4

'o oluo

;g to tustrliltoutoeuoU o st d) puo 'utotuop o s.t ("g)dt - O ueqJ ' iy olu! uE

to uo4catut snonur?uo? p eg uE {_ ull : 6 7a7 'onl uDlI ssq lou ta,a\ut. uo

eq u pI (sureurop Jo acuBrJBlur uo uraroaql s6rar*no.rg) 'tt'B uraroatlJ

L9turppaqurg uralx-a{f,rrJ't't

68 3. Hyperbolic Geometry and Fenchel-Nielsen Coordinates

closed geodesic which is freely homotopic to the simple curve obtained from 4!by applying the Dehn twist with respect to,ti (see Fig.3.10).

F ig .3 .10 .

For every t e Fc,let [ it1,&] b" the corresponding point of ?0. For everyclosed geodesic L on R, we express as I(t) the corresponding closed geodesic onft1, and denote by l(L(t)) the hyperbolic length of I(t). Set

t i( t) = t(Lj(t)), tuo_"+i(t) = t(al(t)), tao-a+i(t) = t(Aj(t))

for every j, and set

I1 t1 = ( r ( t ) , . . . , /go-s( r ) ) .

We have the following:

Theorem 3.L2. The mapping L it o proper embedding of Fo inlo (n+;se -s.

(That is, I is a homeomorphism onto the irnage t(F), and the preimage of any

compact sel in (R+)ee-s under L is compact.)

To prove this theorem, first we fix a point ts of Fs arbitrarily, and write

f r 1 t o 1 = ( o r , . . . , e 3 s - J , e r t . . . , o e g - s ) € ( R + ; a s - t x R 3 c - 3 .

Fix j, and for every s € R, define a point t(s) of d by

t ( s ) = f r - ' ( o r , " ' , a 3 s - s , Q r t " ' , a ! - r t o i + s ' o . i + r , " ' , a s c - e ) .

Then we have the following Proposition.

(Pi . r : Pi ,z)

alsq era acueg 'f1a,rr1cadse.r',? pue og ol ,z pue 0z spues pue '? slt<e eq? qll,r{ luetuele crloq.redfq e s rLoeLuorltsoduror aql uaqf 'slurod paxg se'f1a^rrrleadse.r'0rn pue oz er.ie-q qcrrl^it, olrr1raPro Jo {H)?nV Jo slueruala arldqla eql eq zl pue Il, }al

'pueq reqto aql uO

'(,V' ,m)d | (,m' ,z)d ) (,7' ,z)A

teql r"elc s! ll prr€ '(,n',?)d - (n'z)d 'suorlrugap eql ,(g

.II'8'IIJ

('0 = or leql aunss€ puts {srp }run aql fq p aceldar era eraq^a,11.g.3rg aag)'(rtrlatuoa3 cqoq.radfq erll Jo asues eql W) orn o1 lcadsar qlral ,f1e,rr1cadse.l ,02

pue',2'z o1 crrlatutufs slutod aQ 0z pue '?',V Ie.I

'0o1 pu€ oz r{Snorqt Surssed guo crsapoe3 aqt aq I Ie.I'F f z ueqll. dlqenbeur aq? ilroqs ol seclgns 11'loo-t4

'tm=ffLpuo F

- z lt fi1uo puD fi spyoy frp1onba ay1t.taaoa.to141'fi4ao4ceilsat '(acuoTstp cqoqtadf,tl ayl o7 Tcadsa"t

\lln) p puo m to puo F puo z to sTutod-aypptut ?Ul ̂tD om pup oz ?reqn 'splo1

(m'z)d * (,m' ,z)d) (on'o116

frTqonbaut ?Ul'zI ) ,01,'tn fr.taaa puo rI ) ,z'z fi,.taae.tot'frpualoatnbg '{"1> m'r7 ) z I zC) (n'z)} uo &?auo? fi17cVr1s sr (m,z)d uat17'uaa$ aq H q zI puo rI eusepoa| yu.to[spp QpnTnut onl prl .7I.g BurrrraT

'ur.ro; SurmolloJ eqt ur pe?"ls sr urn+ ur qcrqaa 'd

ecu"lsrp ?Jecurod eq1 ;o {1xa,ruoc Eursn fq pa,rr.rap aq ileqs uorlrsodo.rd srq;

'urnunutur sp sulnllo { t1cryn 7o s

to aqoa anbtun o fl ?r?Vl '.r,o1nc4,toil uI '(E o7 taulilout o so) g uo ..tailo..til puouauor fi17cg"r1s q (((s)l)jill = G)t uotTcun! aatTtsoil ?ttJ .tT.e uog11sodo.r4

"I

m

om

tutppaqurg u-ralx-a{rlrd't't

70 3. Hyperbolic Geometry and Fenchel-Nielsen Coordinates

2p(zs, ws) = p(zo, 2s) < p(z', 2').

2p(ro,*o) 1 p(r ' , w') + p(z,w).

Thus we obtain

tr

Proof of Proposition 3.19.Weconsider the case where P1,r * Pi,, and ^4! inter-

sects ,Li at two points. The other cines can be treated similarly.

Let l's be the Fuchsian group represented by ts, and 7i be an element of l-e

which covers Li(to). On the axis,4; of 1, fix a point zs which is projected to an

intersection point, say p, of Li(t6) and al(to).Let 6l be the element of l-o which

covers Al(til and whose axis passes thiough zs, ar'd Bi be the ax-is of 6f .By

the assumption, the projection of the geodesic .I contained in Bi and connecting

zs to zto = 6l (zo) should intersect Li(to) at some point g other than p. Let z1 be

the l ift of qbn 1, a.nd,4j be the l ift of I i(ts) passing through z1 (see Fig.3.12).

F ig .3 .12 .

For any point z on an oriented geodesic -t

be the point on tr obtained by translating z inon I/ and any a € R", let z(a)rthe positive direction along .L bY

Eurdderu € eleq elvr ueql ', o1Surpuodsa.r.ror tgr uo sselc fdoloruoq ea+ eql ul t, crsapoaS anbrun oql Jo (t2)lr{l3ue1 crloqrad{q aql aq (d!)-l }el

'S I g {raaa rog '1 dq peluaserder ec€Jrnsuueutrrarll eq? eq ,A tel'6,4:l l r(rar'a Jod '(6

{) f snue3Jo Ar er"Jrns pesolc € uoselrn? pasolc elduns Jo sassel? ,fdolouoq ae.r; II€ Jo Surlsrsuoc 1as a{} g ,(q alouaq'srrlolloJ sp pegrpour q U I'g rueroaql ur sp euo qcns t1r;o Surppequra ue '1srrg

'[OZ-V] nreueod pue qcequepn"T'lq1€d ul 'acuelsur ro; 'punoy eq ueo qcrqrrr 's;oord

1p lsourp lpro all 'uolsrnrlf'A{ o? enp sr qclq^a '(e

?) d snua3 3o tg aceds rallnurrlrrel eq} roJ f.repunoq

(te"pt) InJasn e Jo uorl?nJlsuoc aql Jo auqlno ue aar8 llsr{s a^\ 'uorlcas slq} uI

uollBrullredruoc s6uolsrnqJ .7.s

! 'gl't tueroeql uroq s^rolloJ uorlresss eq1 'drc.r1rqre u_f aourg'a(+tI) olur U Jo

Surppaqura radord e sr lr acueg 'snonurluo) pue 'aarlcafur 'redo.rd sr zll otur ?I Jo

("2 + s)rr'(s)rr) --+ s

Surdderu aqt'tI't uorysodor4 fq 11 uo.redo.rd pue xeluor.{11cr.r1s sr (s)/ eeurg

'@z+ s)/ = (((16 +s)i[v)t = (((")r)fv)a

e^€q e1'r 'f, .,(.ra,ra ry (7)17 Euole 1srm1 uqeq eqt Surfldde dq (l)jy uo.rypaurelqo e^rn, eql o1 crdolouroq flaery q (l){V &urs'AI'6 uruoeqJ lo !oo.r,4

tr 'uorlress€ eql aPnlcuo) a^{ 'eroJeq peuorsuetu sB(g)- = (s)rf acurg '11 uo .rado.rd pue xaruoc ,{11cr,r1s st (g)u, ?eql ^\oqs o1 f,seasr 1r

'arourJeqlrng '1urod auo fllcexa te (g)u/ urnturunu eq? su.rep€ ll 'relncrlred

uI'l,V y ly uo rado.rd pu" xaluo? rf11cr.l1s sl (g'rn'r),I'g paxg,{ue roJ snql'g yo drepunoq

aql o1 spuel ol ro z raqlra s" oo+ - (rr"'z)d Wql tceJ aql ruorJ uaes.{lsea equ€f, s€ 'rado.rd sr g 'os1y 'xeluoc ,t11crr1s $ J teql aas uec ar\{ 'tl't eurural ,{g

. (rrvrlels;12)je ,.) a q (,v1s-1m,2)d - (q,^'r),4

3ur11as ,tq U x |V * lVuo (s'rn'z)g uorlcun; ? eugap alr 'acue11

. {(trrrfrls) 127[s,iv(s1n)o *@,,y]i"iu't"' = (r)!

leql aas ue? a/r{

'g = s((07)t7)7!3 "r

{fua1 crloqladfq fq (01)f7 3uo1e 3ur1sr,r,r1 ,(q 0? fq palueserder e?eJrnseql ruo.t; peur€lqo er€Jrns pe{reur aq1 sluasardar (s)1 acurs 'la q13ua1 cqoqraddq

TLuorleryrpeduroC s.uolsrnqJ'?'t

72 3. Hyperbolic Geometry and Fenchel-Nielsen Coordinates

/ . ( 1 ) : S - R *

for every t e Fs, or equivalently, we have a mapping

[: Fo -'--* (R+)t

of ,F'o into (R+)s. Ilence Theorem 3.12 implies the following:

Corollary. The mapping I*: Fo - (R+)s is a proper embedding.

Moreover, this mapping remains an embedding' even when we take the quo-

tient space P(n+;s = (R+)s/R+ as the target space. In fact, letting Pl* be

the composed mapping of /* with the projection n of (R+)t into the projective

space P(R+)s, we have the following theorem.

Theorem 3.15. (cf. [A-29], Expos6 7) The mapping Pl, : Fo - P(R+)s is

an ernbedding.

Next, there is another natural mapping of 5 into (F;s, which is defined by

using the geometric intersection numbers of curves. Here and in the remainder

of this section, we set [T = {o € R I e 2 0}.For any two o1 and o2 € S, the geometric interseclion number f(o1,42) of

a1 and o2 is, by definition, the infimum of the number of intersection points

of .L1 and tr2, where ,Li moves in the free homotopy cla^ss of ai for each j. In

particular, i(ay,a2) = i(oz,ar). 4lro, note that f(o'o) = 0 for every o € S.

Define a mapping i* :.9 -* (F)" - {0} bV setting

i * ( o X . ) - i ( o , . ) , o € S ,

where we denote {0}s simply by 0. Then we can show (cf. [A-29], Expos6 3) that

Pi* - zr o i* ;.9 -----* P(F)s

" i:3:""T: that i* is extended to a mapping of R+ x 5 into (R+)t - {0} bv

settingi - ( c , a ) ( . ) - a . i ( a , - )

for every (o, a) € R+ x S. It is clear that

" (am;O- {o}) = PLIs).

Now, we know the following:

Theorem 3.16. (cf. [A-29], Expos6 4) The subset Fr](S o/P(R+)s is home-omorphic b SGs-7 - {s € R6s-6 | lol = l}.

Remark. The set I.IFD - {0} can be identified with (and is written here-after as) the set Mf of all Whitehead equivalence classes (or more precisely,

0 'uorlJesse aq1 saqdurrqarqn 'r > (l)rt leql qcns ? a rnc pasolo alduns € eleq e^\ acuaH ',|

1e spua pueq}-I/'a stlsls qcrq^r, Jr Jo J3el 3 Jo cr€qns e sr areql 'uraroaql ecuerrncer s<?Jscurod,(B'r > (L)rl ryq1 qcns I cr€ l"srelsrrerl e a{el'0 < I frane.ro;'t:eIul'JW>(rl',1)-dfra.rlaroJ0otBurEra,ruocecuanbasesureluoc{S>ol["]rl]]esaql+eql aoqs u?c e^{ 'uraroaql acuerrncer s(gr"curod Sursn ,(q (pu€q Jaqlo eq} uO

(3 uo Sutpuedap) luelsuoc e,ttlrsod efq rrolaq uorJ pepunoq ars (S 3 o) (o)(l)-/ sanlel aql 't4 3 3 {ra,re rcg.too.r4

'"(tg) ul lW u-ro.{ 7arctnp q (6[).1a6our ay1 'zr'g uor+rsodor4

'g-rsll o1 atqdlouroeuoq eq ol uaou{ $ urnl uI q?Iq^,r'(e.rn1cnr1s xalduroc ua,rrE aq1 o1 lcedsar q?l,rl) U uo sl€rlueraJrp crle.rpenb erqd-rourolorlJo aceds eq1 ql!/{ pagrluepr aq rrcc {0}nJW'g uo arnlcn.rls xaldurocua,rr3 fue .rog 'pq1 pe^roqs [191] lnsery pus preqqnH 'pueq reqlo eqt uO

'lfutv4l - rt

arns?eru l"sralsu"rl lecruouec aql qlpt paddrnba '(7

raldeq3Jo U $ 'Jc) e,rrlrsod fl U uo d prlua.legrp crle.rpenb crqd.rouroloq orez-uou

paqtrasard e qrrqra Jo J"?l f.raaa Euop 'uorleqo; e $ 'arnlcnrls xalduroc e qty(paddtnbe fl U uaq^{ 'g uo uorlerloJ parns?eu e yo aldurexa pcrdr(1 v

'A IJoueA

' lW ) (rt',i) = d leraua3 e rog

ua,ra (g ) d) @)(d)., se uatlrra{ s\ [d]d 'uo ereq uroq{ '(7)r/ d)'Ilur = ldlrl areq^r

'g>d '@)("'D).!-ldlrl

uollelloJ perns€eru " qlr^ir pegluepr q (s 3 p,+lr f p) (,,,r;-, .rffi.t:;y;f]

'g to1 rl arn6?er.u l"sraAsrr"rl e pue 'sarlrreln3urs pelslo$ q?nsqll^ar U uo dr uorl€rloJ € Jo .rred e 'uorlrugap ,fq 'q (r/'d) uo4otlol p?Jnsoeu y

'(!)d = (n)r1 uaql ,4 yo

;ea1 a13urs € ur paurcluoc sr qcrqar Jo llqro qtea 'ddolosr ue fq ,/ euo Jerllou€ olpe^oru aq u3? u - h'0] , la cJ? lesre^susrl e;1 '1eql sauslles qclq/'\ dr Jo sJ€al

Jo scJ"qns lesrelsuerl II€ Jo las aql uo arnsseru e sl 3, .ro; r/ e.rnseatu lesJelsu€r?e '1xa11 '0 = z .{lrreln3urs eq} r"eu '1 re3elur a,rrlrsod etuos q}ua errllrsod sr

zzp,tz q)qM Jo Jeel ,f.razra 3uo1e '{O} - C uo uorlelloJ e Jo teq} s? eJn}cnJ}selqellueraJlp l€ool atu€s eql e eq ppoqs qolq^rJo qcea 'sarlr.repSurs pelelosr qlr^r

A' uo uo.rlsgoJ e eq d la1 '(g gsodxg '[OZ-V] ';c) g eceyrns lceduroc e uo uorlerloJ

Perns"au s Jo uorlluuep aql lle?er e^\ 'e?uarue^uoc Jo e{es eql roJ 'ereH

'Gw)" -- Jwdse (S)-la ssardxa oE" uec e^{ ecueg ('g gsodxg'[OZ-V] aes)'U uo suorlsrloJparnseatu;o (suorleredo s,peaqalrq1\ pu€ fdolosr repun sesselc ecuep,rrnba

TLuorleogrlaeduroC s.uolsrnqJ't't

74 3. Hyperbolic Geometry and Fenchel-Nielsen Coordinates

Now, the crucial fact to construct Thurston's compactification is the followingtheorem.

Theorem 3.18. (cf. [A-29], Exposd 8) For eaery system L = {Lj}?s-13 of d'e-

composing curues of R, there is a nalural homeomorphism

qx : l * (F. ) - . U(L) C Mf ,

where

U ( L ) = { p = ( F , p , ) e M F ( c ( R + ) " ) l p ( L i ) > 0 f o r e v e r y L i e L } .

The construction of qa is as follows. Here and in the remainder of this dis-

cussion, we use the same notation both for a cutve and for the free homotopy

class of it. For every t € Fs, we can construct a measured foliation Pt = (fi,p'r1

such that F1 is transversal to Li for every j, and that

l , ( t ) ( L j ) = i . ( p 1 ) ( L 1 ) , i = 1 , " ' ' 3 s - 3 ,

or equivalently,l ( L j ( t ) ) = t t r l L i l , i = 1 , " ' , 3 9 - 3 ,

where,Ll(f) is the geodesic, on the marked Riemann surface represented by t,

which corresponds to Li as before. We set

q2(1.( t ) ) = Pr '

It is known that this mapping ga is actually a homeomorphism onLo U(L).

Using these natural "projections" qL, *u can derive the following:

Theorem 3.19. (cf. [A-29], Expos6 8) The subsel

Pt . (Fs)U PMT

o/P(FF)s with the relatiue topology is a compacl manifold wilh boundary.

Moreoaer, Pt.(Fs)UPMf is homeomorphic to the real (69-l)-dimensional

closed ball {c € Rog-o I l" l S l}, and the boundary is coincii lent wilh PMf

(which is homeomorphic to Soc-7 ).

we shall show how to construct local coordinates in a neighborhood of an

arbitrary point of the boundary PMf .Fix p[' e PMF and ps e "-t(p[.) arbitrarily. Then there is a system I =

lfij|5" of decomposing curves of ,R such that

f . ( ps ) ( . L t ) > 0 , i = I , " ' , 39 - 3 .

(See [A-29], Expos6 6.) Fromthis 4, we construct afamily {Li,A?,A}}}n=1" "t

simple closed curves which gives an embeddingof Fo into (n+;s0-e as in $3.3.For every e ) 0, set

ad{1 ;o ef,€Jrns uu"rualg e ;o areds rellnuqcrel aq? l€q} ^{ou{ a^it ,.re1ncr1.red

ut ('[t-V] .Uo{lqv 'acuelsur ro; 'aag) '(g)g aceds rellnuqcratr eql yo s3urppequra

s(ulaly-a{?rrd pue sa?surprool uaslarNlaqcuad eqt 'g$ pu" U$ ul suorlrugep oldl.repurrs 'augap uer ear 'arlleur cqoq.radfq aql slFupe U ereJrns e q?ns Jl

'(u'0) adfiy aTru{ fr11ocryfr1ouo

Jo eq o? pres sr (0'u'6) edfl;o aceJrns uueruerg e'.re1ncr1.red uI'(ur'u'0) ailfiy

lo acoltns uuourerq e 'srlsrp pesolc u, pu€ slurod u lurolsrp ,(1en1nur 3ur1a1ap fqf snua3 Jo ar€Jrns uueruerg pasolc s ruoq peur€lqo 'ace;lns uu€tuerll e IIel e \

saloN

tr 'r-o1 tuor; peurclqo sl @Jo asre^ul aqJ'Surddeur-;1es flrll repun ?uerrelur f1.rea1c arc IWd pun (61)*ld'a.reg '"(+U)2,

;o p Surdderu-Jlas snonurluo? s sacnpur ef,uaq pue J1as1r otuo StJo uorlce I€rnleue sa?npur Jleslr o?uo A Jo d ursrqd.rouroatuoq Surrrraserd-uorleluarro fra,rg '{oo.t4

'l1asp oTuo IWd n(6g).ru = (i)*ld lo u.rsnld.tou.toau,oq D saonp-ut t1asyz opo A lo utstrltLtnuoeuoq fun.taserd-uorlDlueuo fi^teag .f.tu11otog

:3ur,uo1o3 aql ,(q u^roqs aq feur uorlecgrlcedruoc $ql yo ecuelrodurl eql'6a p fi"topunoq spolsrn?J pell€c s\ lWd ,(repunoq slr pue '6a

1o uorToc{9ycod-ntoo s.uo?srnqJ peIV) sr tg aceds rallnuqclel erll Jo uorlecgrlceduroe srq;

'od Jo pooqroqq3rau e uI seleurproor l€col se,u3 F

ecuaq pue 'a3eurr slr oluo ursrqdrotuoauoq € sr d Surddeur slql leql a\oqs ue) e^r

'(t'7)p) a 'H+(o)((c)7b).!) (")r > (o)((c)20).r

l€q} qcns y ?uelsuo) e sr eraql'S g ,c ,(rale ro; '1eq1 sa1e1s qcrqu, '.{lqenbaur pluaur€punJ e Sursn fq 'rn.{I

((a)t)d

1nd e,rll, '(t'7)n 3 c ,(rarra .rog

'M)q '(o'd)=(4)o1

se (t'ol x l4 olur (r'f,)2a n /A lool Eurddeur e eugep "'1urf'"j;f""tfi:i[i$sr r, l"ql 3ul1ecag '(ol)Va

Jo ernsop eql ur uado sr (r'l)fla n1t4 lerlt raoqsosl€ uec eM'.1A = (Q'7)n)tb o I pue '(u,I)*ld ur uado s\ Q'J)t)a

'f1rea13'(0)d" - rA pue (("1)n)" = Q'7)nd

'{g-fg, ...,1= ! ,t <(f7)al(u,t).1f c} = (r,7)1

((rtl"lrr 47b)*t+ (ivx(')'b)'t + (D(@)qb)*r) 3 -) o*" '(4'n",)=

saloN

76 3. Hyperbolic Geometry and Fenchel-Nielsen Coordinates

(g,n,^) is homeomorphic to g6s-6*2n*3m. Abo, Thurston's compactificationis considered for such a surface (cf. Fathi, Laudenbach and Po6naru [A-29]).

Relating to $3, we recall the inverse problem, of whether we can determine

a closed Riemann surface by the length spectra, i.e., the set of the hyperboliclengths, of all simple closed geodesics on the surface. This problem is equivalentto the problem of M. Gel'fand: whether a closed surface is determined by theeigenvalue spectra, i.e., the set of all eigenvalues, of the Laplace-Beltrami oper-ator on the surface with respect to the hyperbolic metric. On this problem, seeVenkov [A-110], McKean [152], Sunada [218], Vign6ras 12421, and, Wolpert [246].

For Fricke-Klein's embeddings, we further refer to Keen [110] and Okumura

[173]. The argument in $3 follows that in Fathi, Laudenbach and Pc€naru [A-29],Expos6 7, which is due to A. Douady. As for more advanced investigations on

convexity of geodesic length functions, see Kerckhotr [112] and Wolpert [256].A survey on Thurston's compactification by Thurston himself is given in

Thurston [234]. Fathi, Laudenbach and Po6naru [A-29] is a good introduction to

measured foliations and Thurston's bounda,ry. See also Gardiner [A-34] Chapter

11, Strebel [A-102], Hubbard and Masur [101], Marden and Strebel [137], and

Masur [144].As a generalization of Fenchel-Nielsen deformations, the earthquake defor-

mations have been considered. See Kerckhoff [112] and [114], and Thurston [233].We also cite Bonahon [a5].

Finally, there are many proofs of Teichmiiller's theorem stated in this chapter.For example, see Chapter 5 and Notes of that chapter. There are also proofs by

Wolpert [256] using geodesic length functions, and by Fischer and tomba [74]from a differential geometric viewpoint.

'[g'r] >cfraralsorul".rog [p'a] uo snonurluoc flelnlosqe sr (/i'*)t 'fr

Jo uor]cunJ e s€ pue 'ftt'cl ) fif.ra,ra 1sour1e.rol [g'o] uo snonurluor flalnlosqe x (fi'a)l'c;o uorlcung e sy

:srxe f.reurEeurr eql ol ro srxe I"eJ eqt ol reqlle laleredar" seprs asoq^{ O ul W

'"] x [g'rf = A e13ue1car f.ra,re .rog sploq uorlrpuoc SuraaolloJ eql y (saut1 uo snonurluoc fi1a7n1osqo) nV aq o1 pr"s $ O ur€ruop .reuelde uo (rt'r)l = (z)/ uorlcung e 'araq :lCy q / ter{t Eururnsse ,(q 'eldurexa ro;'paalue.ren3 q s1ql'uretuop pereprsuoc eql q (areqal{rarra lsorule) 'a'e alqerlua-ragrp i(ler1red '1sea1 1e 'aq ppoqs 3f

'luaurarrnber qqt Jo esn€)eg 'esuas alqe?rnsaruos ur ztrl - z1 uorlenbe nueJtleg eql segql"s 1nq'alqerlueresrp fpressacaulou $ qtrqa 3[ ursrqdrouroauoq Sur,,rraserd-uorlelueuo ue reprsuor o? lu"^r eM

sturddel4l luurroJxrootsen$ ;o y uorlrugaq c11,(puy .l.t?

serlredor4 r(.reluawalg pu€ suorl.ruuac 'I'?

'qooq lxal PJsPU€ls uI PunoJ eq lil/'l qcrqrn 'sJoord

1noq1t^/l,p.r3alur anEsaqal ;o froaql aql ul $uaroaql plueurepunJ lereles esn e \

'sJaB '-I pu" sJoJIqy 'rI ol anp 's3urddeur ltsuroJuoersenb roy elnuroJ

Ieuoll"rrel l"lueurepunJ e arrr3 aal 'g uorlcas uI 'uorl"nba lerlua.ragrp rurerllag

erlt Jo uorlnlos aql Jo ruaroeql ecuel$xe eql errord ean '7 uorlcag uI 'suor?ruUep aqlruo.r; fpsea Suurrollog sll€J crs?q elels pue 'suorlrugap esorll Jo ecuele,rrnba aloqs'sSutddeur

leuroJuocrs?nb p.raua3 Jo suor?ruUap IeJaAas alr�3 ea,r '1 uorlceg u1'sur"urop .reueld uaearlaq s3urddeur

IeruroJuocrsenb;o asea ar{t ol sellesrno lcrrlser am 'raldeqc qq} ul 'f13urp.roccy'sur€ruop .reueld uee/\{}aq esoql Jo esec eq} ol sareJrns uustuerg uaarralaq s3urd-deur go esrc aql ecnpar uec e^\ 'uraroaql uorl"zrurroJrun eql fq '1eq1 elop

'aser alqerluere$p eql ur se erues aql sureruar s3urddeur qcns Jo lueul€arlI€tuJoJ ayqal 'suorl"nlrs

IeJeueE arou flqe.raprsuoc w Iool e sr sEurddetu Ietu-roguocrsenb asn ol sn sluolle luauralordurr Iscruqcel " qcns 'uorlruUep eql uluorlrpuos ,(lrpqerluaragrp eql ua{ee^r e,n 'era11 's3urddeu

leruroJuocrstsnb a1qer1-ueJeJIp paugap a^eq ell.r'1 raldeq3 u1 's.raldeqc Ja1eI aql ur pepaeu arc q)Iq^rsEurddeu leuroJuocrsenb;o sarl.radord crseq ureldxa II€qs e^{ '.ra1deqc slq} uI

sturddetr I ler.rrroJuorrsun b

? raldBrlc

?8 4. Quasiconformal Mappings

Remark .1. An absolutely continuous function .F(t) on an interval 1 is differen-tiable at almost every t € /. Hence, when a function f (z) on a domain D is

ACL, the partial derivatives /, and fz of f are well-defined and finite at almost

every z € D. It is not difficult to show that they are measurable.

Now, as a natural generalization of the notion of conformal mappings, wemake the following definition:

Analytic definition A of quasiconformal mappings. Let f(z) be an

orientation-preserving homeomorphism of a planar domain D into the complexplane. We say that f (z) is quasiconformal (qc) on D if / satisfies the following

conditions:

(i) / is ACL on D.(ii) There exists a constant & with 0 < & < I such that

l fd S klf" l a.e. on D.

Setting X = (l+k)/(L-/c), we say that f is K-qc on D. We call the infimumof 1((> 1) such that / is K-qc the marimal dilatation of /, and denote it by I(y

or K(f).

Example -1. A conformal mapping of a domain D is quasiconformal on D. Since

we can take 0 as I in the above condition (ii), a conformal mapping / is l-qc

and K1 - 1.

Example 2. An af f ine mapping f ( r ) = az *bz * c (a,b,c € C, lb l < lo l ) is

quasiconformal. We can tal<e lDl/lol as &, and hence Ks = (al+ l0l)/(ldl- lbl).

Emrnple 3. For a given lc, we set

( z . z e Hf ( t ) =

\ r + i K v , z = x * i y e c - H , K > ! .

Then / is a quasiconformal mapping of C, and Kt = I{.

Remark 2. SetI ( , ) =T :Ep , z€A .

Then / is an orientation-preserving diffeomorphism of the unit disk .4 onto C,

but not quasiconformal. Actually, there are no quasiconformal mappings of 4

onto C. See Proposition 4.32 later in this chapter.

Remark 3. Let f be a K-qc mapping of a domain D onto another domain D',

and g be a conformal mapping of D'. Then the composite mapping I o / is K-qc.

In fact, it is easy to see from the definition that g o / satisfies (i) and (ii), and

that KsoT - I{y.

'g ) n! pue 9 > lrl qlp z fra,re rc1lzlh > (t)tleql osl€ arunss deur errr',,(lessacauJr g rell€urse 3ur1e1 'luarun3re.repturse,,(g'g ) n pue 9 > lrl qtt^ z .,l.ra,re to1 lzll"t > (z)t leql q)ns 9 e,rrlrsod e sr araql

'l((o)u/ - (*)ut)nl +l(s)'tx - (o)/ - (")/l -l@)utn - @)l - @p + n)tl ) Q)1

'l(o)o/n -ft),la -(o)/ -(r){l=(z)t

ef,urs

1eg'dperlrqre I > L I 6 q1r,u l,l xrg'0- 0z ?eq1 eunss€ ar*'flrcqdursJo a{es eq} rod'02 }e elqelluareJrp,tlelol

sl / ?eql ^roqs Iprls e \ pue 'lfi! + oa = 0z lurod " qrns xlJ ('"? ;o sluroddlrsuep eugap e^\ 'fgelnurg 'ofrg

lo uorlcunJ orlsrralr€req, eql fl onal

araq,u

J\ ong

Jo lurod flrsuap e sr 0r 1eq1 ,{es ealr, e.reg) 'flerrrlcedse.r

'{g > fi!+0r I u ) fr) =o"g pue {s ) 0n!+o I u ) n} = ong

slesqns aq1 go slurod ,(lrsuep a.re 0f pue 0r 'g ) ont * 0e dre,ra lsorulero; 'pq1 dldun uraroaql s.lurqng pue rualoeql ,,{lrsuap s,an3saqel '1s.rrg

'f.rerlrqre sr r ro;'g uo'e'e elqetlueragrp ,t1e1o1 sl / leqf ^roqs ol secgns 1r

'uorlresse aq1 a,rord o5'snonurluoc sr ;l' .rog 'g uo

snonurluoc erc nt pue t/ 1eq1 sarldur ecueS.ranuoc ruroJrun aql leql aloN .Ar ueq)ns xU em Joord srqlJo 1sa.r atll uI'0 - q se fl uo.,(1uro;run (r)uI o+ se3ra,ruoc

V I (?) I - @! + z) l) teql etunsse osle {eu aar'1uarun3.re r€lrrurs € /tg'0 1- q e guo flurrogrun (r)'l ol sa3raruoc ,t/(@)l -(rt+t)l) teq? pue r u€q? ssel $ rr-OJo €are er{} }?ql qrns o Jo ar lasqns elqerns"eeu e puu uer e,r,r't}{"6} eeuenbaseql ol ueroeql s,go.ro3g Surflddy 'oo + u se (I uo 'o'e

0 * uf ?sq? aloN

'r - *p(r)oor, ,:*"""'[T

'*#,i"

It't't -lu)q'ulr>lql>oI dns - (r)"0I Q)l - ?t +,)l

1as 'u draae .rog 'f1tre.r1rqr€ ,

1u€lsuoce,r111sod e xld 'eer" alrug e s€q O ueq^\ uorlJass€ eql a,rord o1 sargns ?;''/��oo.t4

'O uo 'e'D alqn4uaufitp fi11o7o7 st t uaqT'g uo 'a'o ut pu, 'l saatToau,ap

1ory.r,od eql sDq C ory! e urDtuop D Io I tustyiltotuoauoq D fi .tV uorlrsodo.r4

'peurtslqo uaeq seq olqarl pueSur.rqag o? enp llnser elqe:lreurar Suro,olloJ eqt 'stusrqdrouroeuroq

Jo ese) aqlur '.rale.tro11 'uorle3rlselur JaqlrnJ o1 alqecrldde ;l';o sarl.radord poo3 aalue.ren3o1 qEnoua 1ou sr a/ pue ,t salrlelrrep lerlred ar{} Jo ef,uelsrxe 'leraue3 u1

sarlrador4 ,(reluauralg pu" suorlrug:aq 'I't

80 4. Quasiconformal Mappings

Further, taking a smaller 6 if necessary, we may assume that, for every z =

a * iy with c, y ) 0 and lzl < 6/2, there exist points x1t r�2, iy1 , and iy2 of Esuch that

Q - r t ) , < n l < e < x2< (1 +4 )c ,

( L - q ) a ( e r < a l U z < ( I + q ) y .

(Note that oo = 0 and ys = 0 a.re density points of Euo and.8ro, respectively.)Then, 1(z*) < qlz* | for every z* on the boundary of the rectangle ,R = lx1,x2)xtyr,yz). On the other hand, the maximal principle holds for /, since / is ahomeomorphism, and hence is an open mapping. Therefore, for some suitable z*on the boundary of .R, we have

I(r) < l f (r.) - /(0) - 0f,(0) - y/y(0)l< I(r ') + l , - r. l ( l / ,(0)l + l /y(0)l) .

Since lz - z*l<r7lzl,we conclude that, for every z with c,y ) 0 and lzl<6/2,

I ( , ) S2q lz l ( r + l / , ( 0 ) l+ l / r (0 ) l ) .

A similar argument shows that the last inequality still holds for every z with

lzl < 612, which implies the total differentiability of f at zs - 0. o

Proposition 4.2. If f is a quasiconformal mapping of o dornain D, then theportial deriaatiues f" and ft arc locally squaTe inlegrable on D.

Prool. First, we have a completely additive set function ,4 by attaching the area

A(E) of f (E) to every Borel subset E of D . Let J y Q) be the density function of

,4 with respect to the twodimensional Lebesgue measure dxdy. By Lebesgue'sdecomposition theorem, we have

(4 .1 )

for every measurable subset E of D.On the other hand, by Proposition 4.1 we see that / is totally differentiable

at almost every z € D, and at such a point z we can show that

JyQ) = l f " ( r ) l ' - l f , ( ' ) l ' .

Since the condition (ii) on / implies that

l fA' sl i l ' s *rt a.e. on D,

the assertion follows by (a.1). tr

Remark y'. Actually, the set function.A in the above proof is absolutely contin-uous, and hence the equality holds in (4.1). See Lemma 4'12 in $1.3.

l"t,p1a,a, s A(E)

el€q a^r uaql '(f)zd(t),d *toJ qlr^1, uollrunJe a{€}'((oo)tl)"$CJo }uetuala ue sv'r o1 leadsar qg,rtr /Jo elrleArrep 1er1.redFuorlnqrrtsrp eql ["/] ,(q aloueq 'W'")xlor'of = (or)U f"r pue ,[g,o] uo 0c

lurod e xrg 'fgrerlrqJe O ur W'rlx [g'r] = g, e13ue1ca.r e xg ,esod.rnd srqt roJ

'1CY

sl ./ l€rl? /roqs ol secgns t! 'y uorlrugep eql Jo esues aql ur leuJoJuocrsenb sr 3l'leql ^\oqs oL' ,V uorlrusap eql Jo esuas eql ul O uo leuroJuocrsenb aq / la1

'/H uorlrugap aql Jo esues eql ur I€ruroJuocrsenb sr

/ 'eaua11 'r(r) uorlrpuoc arll sess-rps y uorlrugep aql Jo esues aq? ur / Surddeur

IeruroJuo?rsenb e 1eq1 t'7 pue 6'p suorlrsodord ss uiroqs fpee.qe a^"q a \ .loo.t4

'Tuapatnba

fr11on7nu.t a.to s0utddou loru.totuoctsonb lo rV puo y suotytu{ag .7.? uraroaql

'O uo'e'€ l'tlq)lttl

?sqlqrns I >{ t0 q?lia { }uelsuocss}sffearaql (rr)'6r uo'f1a,rr1cadsa.r'rt pnt "/ suorlcun; alqer3alur r(1eco1 fq paluasa.rda.r

aq u€? Z pue z o1 laadser qll,lr / Jo salrlelrrep 1er1.red lsuorlnqrrlsrp aql /(l)

:suorllpuoc oall Eur,rnollo; eql segsrles I y O uo lout"loluocrconbsl / leql ,(es e711 'uorl"luarJo se,r.resard q?!q^a C olq O uretuop e yo ursrqdrour-oeuoq s eq / 1a1 .sturddBru

[BturoJuocrsenb Jo ,v uol+rugop c1+rtpuy

:s^{,olloJ qrrqa auo a{t se qcns 'sEurdderu leruroJuocrsenb go y

uoltlugep eqt Jo uorlecglpour e raprsuoc u?c a/r4, 'g'? pue 6'7 suorlrsodo.r4 3ur1o11

s8urddel4l lBruroJuocrsen$ go /I/ uolllugaq c11,tpuv .2.T.?

O 's1rcd ,tq uorler3alur z(q .no11o3suollrasse aqt 'TCV st ;f aaurg 'sp.r3elur paleeder e{l se ue}lrr!\er eq uec (g'p)pue (Z'7)Jo saprs prrerl Ual eql(ureroaql s,lulqnJ pue Z,'V uorlrsodor4 Ag.loo.r,4

(e'r)

(z'v)

'fi,pxpz61 ttil - - fi,papdt 'ttf I

'frprpzdt t"il -=ftpapdt ,t"lI

puD

Toqy snollol 7t 'sq.toililns

Tcndu.toc ypm

0 uo suotl?unt qToorus 11o to 7as eW'(O)JC to dt Tuauala fi.r,aaa .tot 'frputo7J'uorryqpprp

to asuas eUI ur ?soql Vpn lu?pN?uroJ ?JD ,l puo zt saatToauap pty.toil?t#'O urDurop o lo I Futrldout louttotuoctsonb fuaaa JoI .g.V uorlrsoilo.r6

T8sarlrador6 freluauralg pu" suorlrugtag'I'?

82

| | *outt' lte' v)et@)e2(v)dtdv = - | l ro.,

| "" liltr, y) e v (x) da d,v = - I "'

" f (r, v) (p r)' (x) d r dv

4. Quasiconformal Mappings

f @, v) (p r) ' (c) e 2(y) dx dy .

Here, let rp2(y) tend monotonously to the characteristic function of (c' a) (4 €

[c, d]), and we have

fq lso fn lxo

J" J, lf"l(r,y)e{x)dxdv = - J" J, f@,0(pr)'(x)ddv.

Since 4 is arbitrary, we conclude that

for almost every y on [c, dl. Next, for every sufficiently large integer n, take as

gt = gr,n in the above equality a suitable function which is identically 1 on

la*If n,rs-lf nl, and is monotonously increasing and decreasing, respectively,

on [o,o +l/n] and [cq -I/n,rs]. Letting n + oo' then by the above equality,

we have

f to

I lLl@,y)dxdy = f (xo,i l - f @,v) almost every y e1",4. e.4)J a

Here, the exceptional set of y depends on 00. To get rid of this dependence,

consider the set, say -8, of all rational numbers in [a,b]. Since,E is countable,

(4.4) holds a.e. on [c, d] for every o0 in E. since both sides of (a.a) are continuous

with respect to rs, and since .E is dense on [o, D], we conclude that (a.a) holds a.e.

on [c, d] for every cs in [o, b]. Note that for every y where (4.4) holds for every

xo, f(x,y) is absolutely continuous with respect to o, and that [/'] is coincident

with the usual pa,rtial derivative f, a.e. on [o,6].As for the partial derivative of / with respect to y, we can confirm a similar

assertion. Thus we conclude that / is ACL on D and that the distributional

derivatives are coincident with the usual ones. tr

corollary L. Let g be a confonnal mapping of a domain D onto another Dt ,and f be a K-qc mapping of Dt. Then f og is a l{ 'qc mapping of D.

Proof. Let u) = rtr * lu be the variable on D' . By the a.ssumption, there exist the

distributional derivatives /. and /,5, and they are locally integrable on D' .

Since rp o g-1 belongs to Cff(D') for every I € Cf (D), it is easily seen that

there exist the distributional derivatives (/og), and (f "c)r, and that they are

coincident with locally integrable functions

( f . o g ) . g ' a n d ( f * " g ) ' 7 ,

respectively, on D. Moreover, the condition (ii) for / o g clearly holds with the

same /c as in that for f. Thus we have the assertion. tr

Corollary 2. A l-qc mapping is conformal.

'0 - npnp dl"Ut ) - "H)lt [ [ T;lliJJ

t"ql (g'?) 3ur1ou ,tq ,uoqs u?c eilr ecurs pue 'g uo

zt = z(lb) pu€ ,I = "(tb)

acurs 'oo ts u se g uo fprloJrun 3f o1 saSrerr-uoc u/ pue 'u a3rel rflluarcgns fraaa lof (O)JC o1 sEuolaq $

'.reloe.rotr41

(g'f) .t]h) * "dt = z("1) pu€ "(ttt) * "d = "("t)

eleq eilr 'u fleaa roJ uerlJ

')) n 'npxp(z)!(z)t(, -*)"d,"[[ =@)(!u)*udt - (m)u!JJ

Pu€

C)z '(zu)6.ru-(z)"dt

1es 'u .re3alur e,ulrsod fre,re ro; 'reqlrng

vf f '1=npxp(z)dt | |JJ

leql os , lu"lsuo, e esooqC '{srp }run aq1 sr y 'e.rag

les eA\'O vo a7 o1 3uo1eq pue lsrxa

z(lL) pue z(/lr) 'raq1.rnd 'O ul l.roddns lceduroc e seq ll-t uaql 'd Jo poor{roq

-q3reu euros ur 1 o1 lenba {pecrluapr sr qctq^r (O)"$.e ) lr luaurela ue ng'too.t4

(q'')

tDq? puD 'oo + u s0 f uo fiyl^tolmn to7 safi.taauoe "{ p1t Vcns (q)}g u? r-;j{"!} acuanbas D s, areql ,O

lo I lasqnsTaodu.toc tuana.tot ueUJ'CI uo a1qo$ayur fi,11oco7 ato alzllpuD alzllll?t? fils1,7os'fr,pu.rou'e uo aI fr11oco1 ^to zt puD "t senqDtuuap pqlod puorlnquqtp esoqm

O uxprilop n uo uotTcunl snonuNlu@ D ?q t 7a1 'uaafi eq l<dpI .g,V Bururarl

'etutual uorleurrxordde Suurrolloy eql esn e^rdeterqut 'outtual s,fiap1 ;o;oord e elrS e.,n 'acuarualuo?

Jo e{€s aq? ro; ,ere11

D '1fa6 go €rmuel l€crssrlc eql ,iq crqdrouoloq sr 3[ 'ecua11 'uor]nqrJlsrp

Jo esuas eql q O uo 0 - z/ sagsrles O ur€ruop e Jo I Surddeur cb-1 y .too"t4

'v-c)z '0)

v>z ,("+-) dxa e ]=?)d)

's = nprpolz! -,("!)l"lf *tf

'o - npap al"! - ,('!)1"[[ *tf

sarlrador4 freluauralg pu" suorlruyae 'I't

and

4. Quasiconformal Mappings

lim I I l$"), - Qtf)rlo ardy = o,N-6 J JD

we obtain (4.5). tr

Hereafter, we call such a sequence {/.} * in Lemma 4.5 an LP-srnoothingseqaence for / with respect to F.

Lemma 4.6. (Weyl's lemma) Let f be a continuous function on D whosedistribational ileriaatioe f2 is locally integmble on D. If fz =0 in lhe sense of

distributions on D, then f is holomorphic on D.

Proof. Fu a relatively compact subdomain Dr of D arbitrarily, and construct

an .tl-smoothing sequence for / with respect to 4 as in the proof of Lemma

4.5. Flom the construction there, we see thaf (nf)z = 0 in some neighborhood

of E[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[. By (a.6), we see thar (f")7 = 0 on D1 for every sufciently large n. Thus,

/,. is holomorphic on D1 for every sufficiently large n. Since /' converges to /uniformly on I as n --+ oo, / is holomorphic on D1 . Since D1 is arbitrary, we

obtain the assertion.

4.L.3. Geometric Definition G of Quasiconformal Mappings

We give a geometric definition of quasiconformality. For this purpose, we first

introduce the notion of quadrilaterals.

The closure of a domain bounded by a Jordan curve is called a Jordan closed

domain. A, quadrilateralis, by definition, a pair (Q;qt,9z,Qs,Qq) of a Jordan

closed domain Q and four points 8r, Q2, 93, q4 on the boundary 0Q of Q which

are mutually distinct and located in this order with respect to the positive orien-

tation of 0Q. We call each q1 a aertexof the quadrilateral. If there is no confusion,we denote a quadrilateral (Q; er,8z,Qe,qa) simply by Q.

Proposition 4.7. For euery quadrilateral (Q;Cr,Q2,QB,qs), lhere is a homeo'morphism h of Q onto some reclangle R = [0,4] x [0,b] (a,b ) 0) which is

conforntal in the interiorlntQ of Q, and satisfies

h(gr) : 0 , h(q2) = a, h(c") = a* ib, and h(qa) = ib .

Morouer, af b is independent of h.

We call the value alb the module of the quadrilateral (Q;qt,!2,Q3,9+)' and

denote i tby M(Q;8r ,Qz,ee,Qe), or s imply bV M(Q).

Proof. First, by Riemann's mapping theorem, there exists a conformal mappingh1 of Int Q onto the upper half-plane H. By^ Carath6odory's theorem, h1 is

extended to a homeomorphism of Q onto fl U R. Here, by composing a suitable

Miibius transformation, we may assume that

sa{stqos O uroulop o lo { tutililoru cb-51 fr.nog'8'? BurtnaT

'erutuel 3uuro11o; aql e^"q eir ueqtr,'(vb)!' ...'(ID)/ seclllel rllllrl Plel"Urpenb e t" (O)f lePlsuot aru '3 olut @;o/' ursrqdrouroauoq e pu" (tD ,ab'zbtrb 1fl) le.Ialeyrpenb ,(.ra,ra ro; '.rary:earag

tr 'l'? uollls

-odo.r4 Jo uorlress" puoces erll se[du4 qclqa '0 = p pu€ g 4 a ]e{} aPnlf,uoc ert.r

'0 < (zb)y Pu€ '0 < (b)q '0 = (ID)tl = (b)t!

A?UIS

'(O f"I ) z) P * Q)t1c = (z\rt eleq e,rr 'P Pu" t sreqrunu xalduoc elqellns qlr^1'

'acue11 '(c)pV to lueruela u€' o1 papue1xe aq uea , -V o V leqt ees a,u 'flpaleedar

aldrcur.rd uorlcegar (zJs^rq?S Eutflddy '2'7 uotltsodord uI se suolllPuoc aures eql

sessrles qcrqa'r Eurddeu raqlou" aq [g'o] x [g'o] = v

*- @ : q le1 'lxap

.I'?'EIJ

a!T

ffi

x+'l

'(t'f 'EU) Eutddeur P?rrsaP e sI

'b)z 'x*Q)tq"zr1-@)rt

leql ees el'a aruaH .(0 < ,y,x) l,x '0] x [y'>l-]

a13ue1cer auos Jo rolralul eql o?uo g ;o Sutddetu lsruroJuoc 3 sr zV ueqJ

,Qt"rt--lG, -ilzPz

Pue'(sr;tq7l - { ?aS

't < (vb)rq- = (80)It1 pu€ '1 = (zb)tq 'I- = (ID)rtl

sarlrador4,{rcluauralg Pu" suorlruyeq'I'}

s:TI'

"q

<_I't

'H)z| = e)",

98

6 0 4. Quasiconformal Mappings

I

K M ( Q ) < M U @ ) ) < K M ( Q )

for all quadrilalerals Q.

Proof . F ix-mappings h, Q - R - [0 ,o] x [0,6] and [ | f1( - ) - R -

[0,2] x [0,6], presented in Proposition 4.7. Then

F = - h o f o h - r

is a quasiconformal mapping of the interior of R onto the interior of E which

maps 0 , a , i b ,anda* ib to0 ,A , i b , and 6+ iD , respec t i ve l y ( c f . Remark 3 i n $1 .1and Corollary 1 to Theorem 4.4). In particular, .F(z) is ACL on Ii, and hence

for almost every y € [0,D], we have

a < lF(a + iv) - F(iy)l = l,' #o + iv)drl= 1," F"t+ l.zt)dr

Since /J* Jpdrdy S ,4(n) = dD (as was stated in the proof of Proposition 4.2),

integrating both sides of the above inequality over [0,b], we obtain

@b)' < ( [ [ Ur,l+ l+Da"av)- \"r.h " /

- JJ^, 1r1:77d'oa' J Jo," ' '

. [ [ I ( d x d y . [ [ t o o , o y ! K ( a b ) @ b ) .- JJn '

- JJn '

Here, we set l? = { tr , ' e ,R l IF"(*) l I 0}. Thus^we have M(f@)) < KM(Q).Next, replacing f' = h o f o h-r by (ih) o f o (ih)-r (or considering

(Q;qr,qs,q+,qr)), we can show by the same argument that

I K

W@Ds Ma)Thus we have the assertion. tr

Now, by noting Lemma 4.8, we give another definition of quasiconformal

mappings without using partial derivatives.

Geometric definition G of quasiconformal mappings. Let / be a homeo-

morphism of a domain D into C which preserves orientation. We say that / is

quasiconfonnal on D if / satisfies the following condition:

(iii) There is a constant K > 1 such that

MU@)) S IiM(Q)

holds for every quadrilateral Q in D.

'a.rolaq so ftpnp{ilt11 = @)V alaqm 'g lo g ?esqns elglrnsoelrz fri,aaa .ro!

sf f

Q'v) (s)v = nprpQl'll - ,l"lD I Jsa{sr7os CI utvluop D lo I |utddoru Totu.to{uocrsonb fuaag 'ZT'V r.urrrroa

D 'oraz ear? seq ((A)/)r -t = g

les aqt wq+ zI'v eurtuerl {q ureSe ees e/lt '91'7 ureroaqJ Jo (l) fq pu.ro;uocrsenbosle sI r-l eculs 'orez ear€ s€r{ (ar)/ }as eql '71'V eluuo.urc1 }xeu eql dq acuag'A uo'e'e 0 = zt pu€ elqernseeu $ A ueqJ'{0 = "t

I O ) z} = g pg'too"r4

'ouo 'a'o

0+ "l ueql'(I utDu.top o uo lout"toluoctsonb s! I lI 'II'7 uolllsodo.r4

tr ',) uollluuecl ruoq r€al) sI PJIqI eql'(2'7 uorlrsodo.r4 go yoo.rd aq? aag) 'appour ar{} Jo e)u€rJelur leurJoJuoc aql ,(queas sI puo?as aqtr'{reureg Eurpace.rd eq} urorJ s^{olloJ uorlresse lsrg aqa /oo.l2,

'cb-eXrX st t o 6 0utddout pasodruoc eq? '(O)l

{o 6 0utddout cb-zX fi.taaa pun e urotilop D Io I |utildout ab-rX fr".taaa .tog'ab-y oslo s? t_\o I oU 0utddou.t pasodu.t,oc ?q?'e oluo O to t |uzrldout

cb-y fr.tana.tol puo'fr1aai7tadsa.t'e puo q suwu,6p lo tq puo t1 |urildotu

Tout^totuoc fi.taaa .tot 'fi1atuo111 'Tuottiaut fi11ow.to{uoc st fiTtfoutoluoct,sonb-y'cb-y oslo st Outildou cb-y o to as.taaur aytr

'0I'7 urorooql

( '''t ureroaqlo1 1 frego.roC pu€ I'I$ q I {reureg se perlord.{pearle ueaq s€q gl'f ureroeqtrSurmolo; erll Jo (11) wut II"csU)

'6'? ureroaqJ prre 5l uorlrugao Jo sarrsllorot

Ierelas e,rr3 e.r,r. 'a.rag 'y'1$ ol 6'? ruaroeql yo goord aq1 auodlsod geqs all

'guapamba Q7on7

-nur, etD s0utddou.t loru.totuoctsonb lo g puo V suotTru{ap onJ '6'V urarooql

'O ul 0l€replrrpenb .,(.re,ra roJ sploq

@)w>r>(}ilw>@w! I

l€rll qcns I < >I lu€lsuoc e st a.req; ,(ttt)

:euoSurr*o11o; eql ol lualerlrnba fl uorlrugep Suro3e.ro; eql ul ($) uorltpuoC 'qrDuaq

't to t X uorlet"llp lerurxetu eql ol lenbe sr jyr qlns Jo

turuugur aq? 1eql aas a^{'ra1e1 pa,rord $ qrrqa gI'? €ruurarl pue 8'} eurural {g

(t$)

(r)(r)

L8sarlrador4 dreluarualg pu€ suorlluu:aq'I't

88 4. Quasiconformal MaPPings

Prool. First, we consider the case that E is a rectangle contained in D and that

/ is absolutely continuous on the boundary 0E of E.

In this case, in view of Proposition 4.2, we find an tr2-smoothing sequence

{/"}Lpr for / with respect to E' (cf. Lemma 4.5). For every n' put /' = un*�ian.

By Green's formula, we have

t t II I {@^),(an)v - (.,-)r( '")"} dxdv - | u^d'an

J Jn J,E

for every n and. m. Let m * oo. Next, to the right hand side, apply the formula

for integration by parts for the Stieltjes integral. Letting 7r *'+ o9, we obtain

[ [ Uf"P - lf1\ara, = [ [ (u,oc - uuo,)dtdy - [ uda.t t n -

J J n ' - J a n

Here, we write / = u+ia. The right hand side of the above equality is interpreted

as the line integral of uda along the Jordan curve df(E) on the w(- u*ia)-plane.

By the assumption, Af@) is rectifiable. Hence, we can show that

t t l

Ju""o' =

J J,r"rdudo = A(E)'

Thus we have the assertion in the case stated at the beginning of the proof.

Since / is ACL, every recta,ngle contained in D can be approximated by such

rectangles. Hence, (4.7) holds for every rectangle contained in D. By a routine

argument, it is proved that (a.7) holds for every measurable subset E of D. tr

Now, by Propiosition 4.11, for every quasiconformal mapping f of a domain

D, we can consider a quantityf,r t t = T

a.e. on D. This pt b rbounded measurable function on D, and satisfies

ess.sup lptQ)l < 9+ < r.z E D t f y * L

We call pry the compler dilot'ationof f on D.

proposition 4.13. For eoery qaasiconfortnal mopping f and g of o domain D,

the complex dilatation pso!-t of the composed mapping g o I-r is giaen by

. f, Fo -!!_, a.e. on D.l r s o l - r " I = T I _ - t r r l r n (4.8)

proof.By (i) and (iii) of Theorem 4.10, gof-L is quasiconformal on /(D). Hence

by proposition 4.1, g o f-r is totally differentiable on /(D) except for a subset

.O of ,rr"""rrre zero. Applying Lemma 4.12 to the quasiconformal mapping /-1,we see that f-r(E) is also of measure zero. Hence by Proposition 4.1, both /

leqt atunss€ feur aaa '1purs dlluatr-gns & 3ul{"t 'U

}as lceduroc eql uo snonurluoc ,(pr.ro;tun sr 3f acuts 'alo11

('e't 'St,f aeg) 'oo - ll uaqn tlt Aq.{lrlenbeur eq} Jo eprs pueq lq3rr aq1 acelda.r e^\ pue 'llq'lr) = {J 1es ein 'era11

I={'Z/,-!,1?lr-q)-{)13

Pu3

(4+!q)I =2 ,(ofr?+ !e)l =o)

1€rl1 q?ns ({on} x ft)/uo slurod Jo r=l{{)} }es alrug e sr areql ueq; 'f1.re.r}lqre (0 <) r pue f xrg'd1e.rr1cadse.r'({on} x ft)/ pue f7 1o sqfual eqt eq !,1 pu, ll t"j '{ f.rer.a ro;

(A)t = lO pue fft(ofrl x !7 - lg

les pu€ '[g 'r] Jo sle^ralurqns 1u1ofs1p f11en1ntu to r--,I{!il ,tlF tJ ellug e a{el

'6 < & leql erunss? 'uo a.req urorg'on - n - & 1as pue (orf - /t e qcns xlJ'[p'af ) ff frale lsotule l€ alq€ItuereJlp sIg 'uor1cun3 Sursea.rcap-uou e sr (f)g ecurg '[p'a] uo /t f.ra,re .roy ([n'a] x [g'r])/ Joeer€ eql (n).I fq aloua(I'fp,re.ryq.re 0 qW '"1x

[g'rf =A e18ue1cer exr.g'too.t4

'O uo ICV sr g uorTtu{ap aq7

to asuas ?W u? CI utDtuop D lo t |ut,rldout Toutlotuoctsonb finag ''I'' Bruurarl

'(g) p"" (r) suorlrpuoe aql segslles gr uollluuepeql Jo asues aql ur Suldderu leuroJuocrsenb e lsql ^ oqs gI'? pue ?I't s€ururaT3ur,uo11o; eq1 'dlesre,ruoC '($) uorlrpuoc eqt segsltes y uolllugap aql Jo esuasaql q Sutddeu leuroJuoclsenb ts 1eql 8'' eururerl uI u^roqs fpeerp e^eq el6

6'7 tuaroat{I Jo Joord 'v'T'v

tr 'uorlresse aq1 fldurr sarlrpnba e^oqe eql snqtr, 'O J z ,'hala lsorul€ roJ

O*1"^G-t"6) pu€ '0*'6 '0#'l

leqlZl'V evutej pup II'f uo11tsodo.r4 fq ,raoqs uet e^t 'luaurn3re relltuls e 3ursl

'g). lo t(r_/o f) *'l' I o'(,_1o 6) -'0

Pu€

"!' I "t(r-l" 6) +'t' t o ̂G-t o 6) = z6

eAeII a/rr '(r)I - ^Eur1r.r,n 'r{lEurproccy 'p[e^ q elnr urcq? eql (z

lulod e qcns ?V 'O uo z fralelsourle .ro; 'flarrlcadsar '(z)rf ?s pus z le elqellueraslp f1p1o1 a.re 1-./ o d pue

sarlrador4 dreluauralg pu" suorlruyaq'I't

90 4. Quasiconformal Mappings

Ir

r r5 0 > 1

Fi9.4.2.

l f(ro + i(yo +€)) - /("0 +;vil S *for every cs on [o, D] and every { with 0 < € < 4. Take any curve L in Riconnecting two sides of )Ri which a"re parallel to the y-axis. Then we can seethat the length ol f(L) is not less than

I , =i l(* -(*-,1- i .& = 1

On the other hand, let f, be a homeomorphism of some rectangle fr.1 = la1,6il x

l\i,dil onto the quadrilateral Qi which is conformal in the interior of Ei andrespects the vertices suitably. Then we have

/ , 6 ; \ 2 , 6 ;Ii s | |

" tlt'ta,) s tai - ai). |' li, 't 'a,.

\ /a; / t ai

Integrating both sides with respect to y on lei,dil, we obtain

S s u{o i ) .4 i .

Here, we denote by li the area of Qi.Now, suppose that / is K-qc in the sense of the definition G. Since

u(Q)<KM(R)=o+,

(l-) <tV\,,=.ryPT,,

Ir

we get

'ft)"lq j 6)tl 'f11ua1e,rrnbe ro

,$),t - 6)'l -(oETl6)"/ < )/

13ql epnlcuoc a,ll '6 o+ puel r 3ur11a1 'snq6

'(t)o + ffi ? (dDw 7 ('a)wx : x

ul"tqo r '(g'' f '[Og-V] ueuelrr1pue olqerl'ecuelsur ro; '';a) flrpnbaur s,1a3uag fq'acua11 'g?*e = (0)/ eraq!\

' [, ((o)"/ - (o),/) + s's] x P ((o)"/ + (o)"/) + D'Df

elEuelce.r aq1 flaleurxo.rdde sr ('g,)/ .t"ql 'a elrlrsodfrarr.a ro; [r'0] * [r'0] = 'g a13ue1rer e raprsuoC 'ur3rro sqlJo pooqroqq3rau e ur

(lrl)o + z'(il'l + z'(o)"1+ (o)/ = ?)t

se papuedxa q ./ l€rll elou 'asec srql uI

o < (o)"/ teqtaurnsss arrr'acue11 'r?elc $ uorlrassB eql ueql'o = (o)"/JI'(0 <) $)tt 7(il"t'6'7 uorlrsodordJo;oord eq1 ur pelou e \ sy'anrleEau-uou a.re (g)z/ pu€ (0)"/

l€ql eurnssp JeqlrnJ feur an (20 pve Id sraqurnu leer alqelrns qq^a (z . "sre)t . ,6reSutreptsuoc ,(g '0 = oz Ie-qI'flqelaua! Jo ssol lnoqtl^t

'eurnss€ feur er* ara11'oz = z e q?ns xld 'O ) z f.rcrlo lsorul€ le elqerluaragp f11e1o1 sr 3f

'acua11'1'7 uorlrsodord Jo uorldurnsse aql segsrles / teqt saqdur yl't €ruural 'loo.r4

'0+x)10-x)=qer?qn

'O uo 'e'o l'llq > Itll

uayl 'N uor?Dlolrp lourroou eUl ypn g uotTru{ap aqy

{o asuas ?qI u? CI urvulop o to 0utddoru Tou.totuottsonb p sr I lt 'St'V BtutuaT

D 'O uo rIcY sr

/ reqr epnl)uof, ar'l snqtr, '[g'r] > 0c .frerra lsotup ro; d ;o uorlcunJ e se [p'a] uosnonurluoc {1e1n1osqe * (fi'oa)l }€q} ^\oqs uec ea\ 'luaurn3re eures aq1 fg

'o Jo uorlrunJ * * [g'o] uo snonurluoc

flalnlosqe 4(fr'r)!'flluanbesuoC'0 * llt--,t'3se 0 - !,lt--,j31€rll apnlruo?

e.!r', - lA? \e?urs'ellug sr f7 r(.ra,ra l€ql ees rr€c ea\'relncrlred u1 '6 <- & se

anl€ etlug e ol spuet (ofr - q/((on)a - (n)A) '0n te alqeltuereJlP sI dr e)uts

I6sarlrador6,(rcluauralg pup suorlluya('I'p

92 4. Quasiconformal Mappings

4.L.5, Other F\rndamental Properties of Quasiconformal Mappings

We state here, without proofs, two of the fundamental and important propertieson continuity of quasiconformal mappings.

Theorem 4.16. (Mori's theorem ll57D A f is a K-qc mapping of the unitdisk A onto itself with f (0) = 0, then

l f ( r t ) - f ( r z ) l < t 6 l z1 - , r l t lK , 21 ,22 € A , z1 f 22 .

Theorem 4.17. Eaery sequence of K-qc mappings of C onto itself f ir ing 0 andI conlains a subsequence which conaerges unifonnly with respect to the sphericald,istance.

Moreoaer, the l imil function of such a subsequence is again I{-qc.

For proofs of these theorems and further information on quasiconformal map-pings, see, for instance, Ahlfors [A-2], and Lehto and Virtanen [A-69].

4.2. Existence Theorem on Quasiconformal Mappings

We have seen that a quasiconformal mapping / of a domain D induces abounded measurable function pJ on D which satisfies ess.sup362lttt!)l < L.In this section, we shall show the converse. Namely, for every measurable p with€ss.supz(DlpQ)l < 1, we construct a quasiconformal mapping whose complexdilatation is equal to p.

4.2.1. Preliminary Considerations

We denote by I-(D) the complex Banach space of all bounded measurablefunctions on a domain D. Here, the norm is given by

l lpl l- = ess.supz€Dlp!)|, p e L*(D).

Let B(D)1be the unit open ball {p e L*(D) | l lpl l- < 1} of L*(D), and callany element of B(D)1 a Beltrami coeficient on D.

First, we note that a quasiconformal mapping with the prescribed complexdilatation is essentially unique. More precisely, we have the following:

Proposition 4.18. Let pr be an arbitrvry elernent of B(D)1. Supposethatthereerists a quasiconformal mapping f wilh the cornpler dilatation pJ = p. Then foreaery confortnal mapping h of f (D), the cornposed mapping ho f has the samecomplex dilatotion p.

Conaersely, for euery quasiconformal mapping g with Fg = H, the composedmapping g o f-L is a conformal mapping "f f (D).

(o'r)'c r )'(c)az ) tt'op,p (1- +) etu"fl +- = o)qa3ur11es ,tq (C)aZ uo d roleredo reeutl € augap eA\

tr 'uotllesse eql aAeq e^r

'@m e u se 'flartlcadsa.r '(g)a7 ul zt * "("/) pt" 'g uo ttpuroyun / +- u/ aautg

,Pv,Pffi"il+-ffi"f +=e)"!se,rr3 elnurro; s(uearp 'u fla,ra rog ',tpe.r1rqre g ) ) lutod e xg

pu"'g o1 lcadsar qgi'a / roJ I*{"/} aauanbes Sutqloours-o7 ue a4e1 '1xa1i

'flqenbaur s(raploH fq s.uo11o; uorlresse aq? 'g uo alqe.rEalul sI rl() - z)/11 acurg'I = b/I +dlI fq paugep aq (Z >)D lal

'trsJ u1 'alqer3alur f1a1n1osq€ sI apls

pueq lq3rr eql uo turel puo?es arll Jo puer8alut eqt l"ql a?ou erlr '1sq1g /oor2r

'C u! g tlnp uado tueaa .tot

a ) ),nP,pfi'ff +_ #"'[,+= o)/sa{s4ns t uaqJ '(dn Io

tt puo lt s7uau;1a fr,q peTuesaul^t eJo searry)auap 1otq.tod PuorlnquqrP esoqn Cuo uotTeunt cnonutluoz o ?q I pI '@ > d > ?, qfn d qI 'At'V uollrsodorg

'uralqord-g aql e^los ol Frluasse x opru"tot s,ntadtuo4 letrsselc Euu,ro11o; eqa

'(r)ar ),t ot,(np,por:[f)

= oil,ltl

,(q uarrrE fl rurou aql eJeqA{ 'C uo

alqerSelur a.re alll 1etll qcns c uo / suollcunJ elq€rns?au 1e 3o aceds qc€ueg

xalduroc eq+ eq (C) n lel 'oo > d ; I qll^{ d f.ra,ra rog 'uolleruroJsuerl fqcne3

aql m ua\our1 iflecrsselc u e/ urog / lanrlsuooar o1 fem e 'pueq Jaqlo eql uo'uorlenba ruerlleg ua,u3 aql p ((rl)grl)9 = t

uorlnlos e ur?lqo a,r,r,'(r/)g = zt tjd'rol aql u-r 11 3ut1t.ra,rag ''l pue r/ uaaallaq

'(trt)C='el)g=ztuoll€lar e e^sq aru ue{}

'et)C - / uorleluaserdar elqellns e 1eB a,n 3r ',t1en1ry 'zJ' ue^€ eq? urorJ /

elqelrns e Surpug ''a'r 'tualqold-g ..If Eunlos 1srg rePlsuoc ea. 'asodlnd uq1 rog'"lrt - zs

uorlenba 1er1ue-regrp rtu€rlleg eql e^los ol lr.roq replsuoc eaa'I(C)B 3 r/ ua,r€ {ue ro;',tlo1q

tr 't'? ureroeql o1 6 f.re1oro3 ,(q leurro;uoc 8I etueq pue 'rb-1 sr

'-Io61€rll s&olloJ tl'gl't uotltsodor4 fq (O)/ uo'e'e g ol lenba s.r '-lotrl acu-ts'1xatr1 'rl = trl - {ottil e^eq e/{'tI'} uo11rsodor43o;oord aql uI s€'1s.rrg /ool2'

stu-rddery l?uroJuof,rsen$ uo uraroaql eluetsD(g 'Z'?

94 4. Quasiconformal Mappings

Then we have the following:

Lemma 4.2O. For eaery p wi th2 < p < q and for euery h e Lp(C), Phis a uniformly Hii lder continuous function on C, wilh erponenl (7-2/p), andsatisfies Ph(0) = 0.

Moreouer, Pf satisfies(Ph)a = h

on C in lhe sense of distribution.

Prool. First, as in the proof of Proposition 4.19, we shall show that the integralon the right hand side of (4.9) is well-defined. For this purpose, define q by theequation \/p+I/q = 1. Since

1 1 Cz - C - - r = r 1 r - q

belongs to Lq(C), Hcilder's inequality implies that

1 aIPA(C) l< ; l l h l l " l l ; *n l l c ( @.

Further, when ( I 0, by changing the variable, we have

t r t . t t Q , f l l 1 l o ,II l--:--- l axay=lCl2-z'II l--;----l d,rda.

J J c l z ( z - C ) l J J g l z ( z - L ) l

Hence, there is a constant I(o depending only on p such that

lPh(( ) l SKr l lh l lo ' lc l ' - ' to , (€c, e+0. (4 .10)

Since Ph(0) - 0 by the definition, (4.10) is valid even when ( = 0.Next, set hr(r) - h(z + Ct).Then we obtain

phlCz- (,) = -+ il"h(z + c,) (4+_6- l) o,o,1 f f / 1 1 \

z r J J c " \ z - ( z z - Q /= Ph(cz) - Ph((r) .

Combining this with (4.10), we conclude that

lph((r) - pn4)l S Kollhllo ' lc, - crlt- ' to, cr,cz Q c, (4.11)

or equivalently, that Ph is a uniformly Holder continuous function with exponent| - 2/P'

To show the second assertion, take a sequence {h"}Lr in Cf (C) such that

llh - h"llo * 0 as n ---+ oo. (Such a sequence is constructed, for example, as inthe proof of Lemma 4.5.) Then for every h., we have

leql pu€ '(C)"3C ) q ,trerra roJ (C)-C o1 sEuolaq qd IeqI(gt't) p"* (Zt'f),(q aas u?c aa,r'1xa11 '(tt'l) paglre^,(pea.rp e^"q eM'too.t4

(qrr)

ftr'v)

(tn)

lzt'v)

96

'zllqll = zllutll

'C uo rtJ = "(Ud)

puD

sa{s4os (C)JC ) t1 tuaag 'TZ'7 Bururarl

'(c)"3c >,1 '�{**fft*rt-4t11i-}"i = o),tt

1er?e1ur relnEurs eql ,iq

peugap ; roleredo r€eull aql ol uorluelle .rno fed am 'acua11 'lerluasse $ lurel

puores aql '0 e , sB 0 ol sa3rarruoc apls pueq lr{3tl aq1 uo tural }srg aq} eculs

.{ rru,rz) - z) t"tt-"l\[[?"2

+ ro1 ,'t,=l)-zl[ ryg]ni =f.-' (4,1 lJ | -'(z)u I Y)

)-zcffz:t6zPv zPGfi ll ;

= o))('1a)

sarrr3 elntu.ro;

s(ueerC'asec stql uf (C)"3C o1 s3uolaq q eJaq^l esec aql eulurexa aal.'asod.rnd

slql rod '"(qa) roJ uollsluasa.rder 1e.r3a1ul elqsllns e ul€lqo ol Peou ear'1xa11

tr 'uollresse

Puoces aq1 pe,rord e^eq e/rl snqJ

cf f )f f'(c)Jc>a'fipxpz41''ld

I I -=npxpdq | | IJ JJ

seat3 ,(1t1enba a^oqe aql'oo + u lal e^r ueq/t'acue11 '(Ot'l),tq

C (lo slesqns lceduroc fue uo r(luro;tun''a'r) uo .{pr.royrun rtlpcol t12, o1 seS.rar'uot ut14'oo s u se 0 F oll"U - qll ecutg

'frPaPz4''' "'ld"[ [ - - fiPxPdtu;t tJJ JJ

1eB a,ra. '(C)JC lt d .{rerra ro3 'relnctlred u1

.()),q = rpY*{=tt-zt! +t;ri = (illeltd) ' (z)"tl I t

se,rr3 elnurro; s.uaalg ecueg

)-z cff v 'np,pG)1",1 Jlt-=

(^0.r6{;s"ll +)# = eFyta)s8urddel4l l"uroluof,rspn$ uo ruaroaqJ af,ualsrxg 'Z''

96 4. Quasiconformal Mappings

1rh17= -* ll;ro>"r-rolzdz AdZ

= + Il"ph.(-ph),,d2 AdZ

= * ll"ph (h)zdz ^dz

= -+ ll"nenl,dz A d2 = llhll1.

Thus we have proved (4.15). tr

Lemma 4.21 implies, in particular, that the operator 7 is extended to abounded linear operator on L2(C) into itself with norm 1. Since we have con-sidered the operator P as that on LP(C) with p ) 2, we consider ? also assuch an operator on .tr(C). Then we see by the following classical Calder6n-Zygmund's theorem that ? gives a bounded linear operator on U(C) (p > 2)into itself.

Proposition 4.22. (Calder6n and Zygmund) For eaery p with 2<-p 1crc,

Cp - sup ll"hllon€ c8p(c), l la l l r=l

is finite. Hence, the operalor T is edended to a boanded linear operator of Lr (C)into itself with norm Co.

Moreoaer, Co is continaoas wilh respecl lo p. In parlicular, Co satisfies

lim C. = 1.P - 2

(4.16)

In $4, we shall include a proof of this basic result for the sake of conve-nience. Here, assuming this proposition, we solve the Beltrami equation. Notethat Proposition 4.22 gives the following:

Proposition 4.23. For an arbitrarily giaen p (> 2) and euery h e U(C),

(Ph), = Th

on C in the sense of dislribulion.

Proof. Take a sequence {h"}8, in Cff(C) approximating h in Lp(C) (cf. theproof of Lemma 4.20). For every n, (4.14) implies that

f I t lI I r n " .gdxdy= - I I pn " .g ,dady , p€Cf (C) .

J J c J J c

Here, Ph,, - Ph locally uniformly on C by (a.10) and Thn - 11, in 7r(C) bVProposition 4.22, respectively, as rl + oo. Hence, we obtain

epnltuoc a^r eunual s,1fa6 r(q ure3e'ecua11 'C uo'e'e zD = zt salrE osle uorlenberurerlleg eqt uaqJ'C uo'e'" "6 =,! 1eB aar'uorldtunsse aq1 fq 1 ) dg1 ecurg

' oll' 0 -' I llo Cq ) dll(" 6,t),t - $ rt),tll = dll, 6 -, tllul€lqo e$'ZT,'V uorlrsodor6 fg

'l*('6rl)a="0

aA?r.larrr 'arroqe se 'ueq; 'f uorlnlos l€rurou Jeqloue sr areql 1eq1 asoddns 'uorlnlos

Ier.urou eql Jo ssauenbrun er{} ^roqs n"qs eAr '(gf 'f) uorlenba Sursn '1xa11

(st'r) 'l+('tilh= "tuorlenba eql ulelqo

fleug all^'tC'V uorlrsodor4 Eurlou pue z ol lcadsar qlr^{ sa^r}e rrep eql Eur4ea

(trv)'C)z '�z+(z)(zt)d=?)t

ecueq pue'0 = o aA"rI a^{'0 = (0)/ acurg'(3 ) D) D + z - (z)I''"'!'I - (r),,tr

l"eql epnlcuoc usc a^r snql 'I - d saop os '(C)al o1 Euolaq (rt)l = '(t)a

pu€ I - J ecurs (pu"q raqlo eql uO 'C elor{A{ eq} uo crqdrouroloq q (z),9

1eq1 sarldun (9'7 eururel) eurural s,1fa7y1 'acua11 'uorlnqrJlstp eql Jo esues eqlul 0 = cdr rraloerol

tr 'O = (O),9 pu" snonulluoc sr (z)g '96'7 eurural fq ueqa

'c)z '(r)(l)a-Q)t=Q)ttes '(Da reprsuor ue? e,r,r snqJ '(C)d? o1 sEuolaq osle zt '(C)al o1 sEuolaq 1 - f

eculs pu" 'lroddns lceduroc e seq ztd - ?'J' acurs 'i(;sr1es plnoqs rl rc1 I uorlnlos

I€Lurou eql Jo "/ elllellrap 1er1.red aqt qcrr{irr uorlrpuo? € aArrap aaa '1s.ng

/oo.l2'

I*Iluara'rp rurertag aqrrouorwlos purrou eqrv.''iluaroeqtr,r,l rl#rlojii"o"

'suor,ppuo? asaqT frq frqanbtun peurulJepp sy, I uo. qcns 'taaoa.to14J

lrt - z1

'uotlnqu?s'P to esues eql u' J uo

sa{n1os t puo '(3)67 o7sfuo1aq I - 't '0 = (0)/ lDUl Urns t uotTcunl snonuNluo) D slsrr,e anqT 'l,.toildns

Tcoilutoc q?tn puv { > -llt/ll ql4rr" r(C)g ) rt tuaaa .tot uaqa 'I > dCtt UWn

k <) a aarTtsod, o ?tloJ'fr1g.to"r.7gy,n I > { t 0 toql qrns q a!.{'VZ'V urarooql

'tueJoeql lelueuepunJ Eurmolloy aq1 alo.rd o1 {pea.r are alr 'mo1q

suol+nlos [BurroN aql Jo acualsrxg 'z'z',

'(c)Jc r a'ftprpzdt,""[ f

- - fipxpdt,t"il

L6sturddul4l l"ruroJuorrs"nS uo uraroeqJ af,ualsFg 'Z't

98 4. Quasiconformal Mappings

tha t / - gand l - g u r . ho lomorph i con C , wh i ch in tu rn imp l i es t ha t f - g

should be a constant. Since /(0) = S(0) = 0, we conclude that / -

9, whichimplies the uniqueness of the normal solution.

Finally, the existence of the normal solution follows also from (4.18). In fact,repeat substituting the whole right hand side for /, on the right hand side. Then,we have the following formal series for f" - l:

f" - | = Tp * TjtrD + T(p,r@Tp)) + . .. .

This series actually converges in ,Lp(C), since the linear operator which sendsh € Lp(C) to T(p.h) € ,p(C) has the operator norm not greater than &Co (< 1).We set

h = T p + r f u r f i + . . . .Then h belongs to trp(C). We shall show that

(4 .1e)

f ( ' ) = P ( p ( h + I ) ) ( z ) + z

is a desired solution.In fact, tt(h+ 1) belongs b Lr(C), for p has a compact support. Hence,

Lemma4.20 implies that / is continuous, "f(0) = 0, and 7, = p(h-t 1). Moreover,Proposition 4.23 irnplies that

f " = T ( p ( h + 1 ) ) + 1 = h + 1 .

Hence, / satisfies the Beltrami equation fz = pf,, and f" - 1 belongs to Ip(C).!

4.2.3. Basic Properties of Normal Solutions

From the construction of the normal solution in the proof of Theorem 4.24, wehave immediately the following:

Corollary l, Underlhe same circumstances as in Theorem /.2l, lhe followinginequalit ies hold:

(4.20)

and,

l/((r) - f&z)l s fi^lrllpl(r - czlt-ztp+ l(r - (zl (4.2r)

for euery Cr,Cz €C. Herv, I{o is the conslanl giaen in Lhe proof of Lemma y'.20.

Proof. Let h be as in the proof of Theorem 4.24. Since h = T(prh)*Tpby (4.19),we have

llhllo S kcpllhllp + collpllo.

Since f/; = p(h * 1), we obtain (4.20).Next, by (4.17) we have

Ill|illp S * llpllp,

L - / I ; V p

'(c uo'a'o) rl - trt sa{sqos puo'5to |urddou Tottt"totuoetsonb o st rl .tol I uo\nlos

IDttltou eW 'fA'f uteroeyJ u, sD snuolsurnrrn eutDs eV? repun .gU.' uraJoaqJ

'3;o Surddeu leuroJuocrsenb e sr erueq pue 'g1as1r oluo C Jorusrqdrouroauoq e '1ce; u1 'sr uorlnlos l€urrou eqt teql ^\oqs ileqs e,u, ,inolq

tr 'g uo fprro;tun / o1 sa3ra,ruor Y fnqfepnlf,uoc aan 'u {ra.re roJ oo Jo pooqloqqftau paxg e ur crqdrouroloq sr ,f - $ acurg'oo <- u se C uo rlprroyun f1eco1 t * "! lsql aes eilr snqJ .C I ) {ra,le ro;

* -rl)l {dll'( "l) - "Illt + dll'�l(,t - ,t)ll} ox }

l()X'(V) -'l)al = l())Y - O)/lurclqo e,rn '(0I'?) pue (ZI't) fq 'txaN

'(ZZ'V) 1aB a,r,r, 'oo * s se 3 uo 'a'e r/o1 seS.ra,ruoc url ecurs pue 'pepunoq ,tpr.rogrun etr- url 1e yo slroddns eql ecurs

.dllz!(uil - ql:"T; t, dll,('!) - "!ll

al"q ell 'acue11

'dllzl(d -,t)lloc + dll'("1) -'lllocq )oll("led -,t)).r,ll + oll(('(V) - "t)",t)l;ll, oll'el) - "!ll

kz'v)

selr3 (91'7) 'u .,(.rarre ro3 '1srrg 3foo.r4'

'0 = dll,! - "(Y)ll ?i,r"

puo 'a + u so 3 uo fiTnttotr,un t +- uI ueqJ. rt t o t u o,4n r o s * *

:, :, :, "r :" : " ::,, :;" ;r: : ::," : r,:^::," :, r.J

rr,tpuo 'u

to Tuapuadapu! W luolsuo)elqo?rns D y?!n {W > lrl I C ) z} u? pauwtuo? T"toddns o soy url tuaaa (n)

'u tuaaa rol tt > *ll"rlll (t)

:suot?xpuo? 6utmo17ol aq7 0utfitstTos r(C)gur acuanbas o aq r/{ur'} nf 'fa'f uero?qJ u, sD eq d puo t1 pI .Z it.relloaog

'sllrolloJ s" sluarrlgeocrtupJ]leg eq] uo suorlnlos leturou eql Jo ecuapuedap eleq a^\ 'arourJeq]rnd

! '$6V) pu" (II'p) fq s,rao11o3 (16'y) acueg

'lz) - r)l +l(z))('l)a - ())el)dl t l(z))/ - (r))/l

s8urddey4i l"urroluof,rssn$ uo uraroaql af,ualsrxg 'Z't

100 4. Quasiconformal Mappings

To prove this theorem, we need the following generalization of Weyl's lemma(Lemma 4.6).

Lemrna 4.26. Let u and o be continuous functions on a simplg connecleildomain D whose distributional padial derioatiaes can be reprc.sented, by locallyintegrable functions. FurTher, suppose that u2 = u". Then there edsts a funclionf which is continuously differcntiable (i.e., of class Cr ) and satisfies fz = u anil

f r = u .

Proof. Ftx a rectangle R in D a^rbitra"rily. Take an .Ll-smoothing sequence

{,rr}Lt and {or}f;=, on rR for u and o, respectively, with respect to r? suchthat (u")7 = (u,)" for every n. (This is possible. See the proof of Lemma 4.5.)Then for every n, Green's formula gives

{("") t - (o^),} dz A dz - 0.

Hence, letting n --+ oo, we have

I pa, * udz) = o.J 8 R

Since ,R is arbitrary, we conclude that the indefinite integral of udz * udZ iswell-defined, and gives a desired function. tr

Returning to the proof of Theorem 4.25, we take a sufficiently large M sothat {z e C I lzl < M} contains the support of p, and fix it in the rest of thisproof. Fix also a sequence {p"}flr in Cf (C) with llp"ll- ( * such that thesupport of p, is contained in {z € C I l"l < M} for every n, and that p.n - 1ta.e. on C as n --* oo. We denote by /. the normal solution for ptn for every n.

Lemrra 4.27. In the foregoing situalion, fr: C ------ C is a honteornorphismbelonging to Cr(C) for euery n. In particular, euery fn is quasiconformal.

Proof. Consider a function g with 9z - pngz. Set

tt = gz and a = gz = I,tnt-r,

To show that 9 belongs to C1(C), it suffices by Lemma 4.26 to see that u isa continuous function which satisfies u7 = (lt"u)r.If we set a = logu, this isequivalent to proving that o is a continuous function which satisfies

c z = F n c z * ( t t " ) " . (4.23)

Now, differential equation (4.23) is solved in a similar way to the case ofthe Beltrami equation. In fact, first as in the case of (4.18), we can construct asolution h in IP(C) of the equation

[ 6^ar*u^dz)= [ [JAR J JR

i , =r1p^rt1+T((p),).

'C ) eztrz frreae rot

l(zr)"1 - (r)"ll +

a1"-rl(zr)'! - (tz),!111'r11ft +r(4 - t) ) pz - r71 ftz'v)

sa{s4os ut fuaag '62'} BrrruraT

:3uraro11og eql pe?u e,u 'arourraqlrng

tr 'ursrqdroruoauroq

" q./'snql ''C eloq!\ eql uo qeu"rq panle^ e13urs e *{ r_/ pq1 serldurruraroaql {ruo.rpoirour l"rrssrlt aq1 'peleauuoc fldurrs sr m3 acurg 'nC uo eAJn?fraaa 3uo1e flecrlfpue panurluoc eq uec rl Jo qcuerq fue pue 'mg

;o 1u1odfue;o pooq.roqqSrau e ur seqf,u?rq arqd.rouoloq wq r-.f

'uolldurnsse fq mop('crqd.rouroeuroq f1pco1 s1 ,f roy 'pa8ueqcun sr z3 uo f3o1odo1 aq1

1eq1a1op) '1 raldeq3Jo I'7'I$ ul s€ '? uo ernlcnrls "q1 1c*q 3uq1nd fq 'C uoernlcnrls xalduoc alau e Surcnporlur ,tci auop aq uec stlJ 'uorlcun; crqdrourbloqe se / raprsuoc o1 sr ursrqdrouroeuoq € sl / ltsqt aoqs o1 ferrr auo '1xa11

''? = (e)t 1eql apnpuoc a,ra'(,p)/ sr os o?uaq pue'lcedtuoc sl "?

ecurg'uado q ("?)/'Surddeur uado ue sr 3f acurg'flarrrlcadsal'/;o 1aEle1 aqlpus ulsuutop eql ere qcrqn sareqds uu€tuet1l aqt 'C,tq pue '9 fq eloueq'too.t4

'C oluo g lo tus.tyd.touro?uto! o fi11onycosr t uayT 'crqtuoruoatuotl fi11oco1 sl ?

<- g :i uotTcunl o fi 'gZ'7 BururaT

! 'Jleslr oluo 3 go ruslqdrouoeuoq s sl V 1eq1 salldun eurual lxau aql

'aro;e.reqa 'C eloq^{ aql uo crqdrouroaruoq f11eco1 osl€ sr ut (@ = z 1e alod aldurrse wq V acrirg '3 uo crqdrouroeuroq flecol q V

'ereq,nfterla earlrsod sl V Jo

"eel. (el"dl - r) = "lr(I)l - "l"et)luerqocef eql ef,urs 'rl11eurg

'1c sselcJo sl "/ t€q? apnlcuor ellr snqJ,'"1 - 6 +eqlserldtur uollnlos leturou eqlJo ssauenbrun aql'acua11 '(C)al o1 s3uoleq I- r0

leql aes aal 'g 3o ecroqc eq1 ,(q 1 = @)t

**"u41 pu€ oo - z lo pooq.roqq3raue ur crqdrouroloq sr t = zD aours '0 = (0)t l"ql ewnss? feru arn 'ara11

'(rurl -1z6url - z6 Surfgsrles (C)rC ] f uorlcun; e slsrxa ereqt leql sarldurrgA't surua'I snql '"(t"rl) - zt Pue snonulluoc sI r uaql '

oe = ! 1as 'u,o11

'GZ'V) Jo uollnlos snonul?uoc e flenlce sL slql

'"("rt) * qurl = zo pu€ rl = ((rl) * qurl)a = ,o

"".rrrl'oo = z Jopooq.roqq3raue ur crqdrouroloq sr r leql aloN)'0 = (z)o -*"urll l€ql os C esooqc alu\ ereg

'C+(eil)*q"rt)4 - o1as'1xep

TOIs8urddell FruroJuof,rsen$ uo uraroaql af,ualsrxg 'Z't

702 4. Quasiconformal Mappings

Proof. For a fixed n, we can see from the proof of Lemma 4.27 that (/,)-1 is ofclass Cl. Also, (/,)-r is quasiconformal by (i) in Theorem 4.10. By Proposition4.13, we obtain

t t ( J . ) - , o f n = - * ' r , .lh) '

In particular, putting un = p(I^)-t, we have lr"" f"l = ltr"l (a.e.). Thus we have

t f l l| | lv"lpdxdy = I I lp"lo(l(f")"1' - l(f"),12)axay

J J C , J J C

t [ | l , , l , l ( f^)" l2drdy = [ [ U,^l '- ' l ( f^)zl2drdyJ J c J J C

/ r t \ ' t ' / r f \ ;= (//" lu"lP dadv

) \J J"l$"),lP dxdv

)

Then (4.20) in Corollary 1 gives

l l r" l lo S 0 - kc)-3 . l le" l lo.Thus, applying (4.21) with / = (.f")-t and (i = f"Qi) (, = 1,2), we con-

clude the assertion. !

Proof of Theorem 4.25.Let {p"}T=t and {/,}Lr be as defined before Lemma

4.27 . Then /,. converges to / uniformly on C by Corollary 2 to Theorem 4.24.

Since l lp"l lo * l lpllp (n --* oo), $.2\ in Lemma 4.29 is sti l l valid when we

r^eplace /, and pnby f and, p, respectively. Hence we conclude that /'0 -----

C is a continuous bijection, and therefore is a homeomorphism. Next, since

f , - | belongs to Lp (C), so does f z = Ff ,. Thus, / satisfies the assumptions inDefinition At, and hence is quasiconformal. tr

4.2.4. Existence Theorem

We have shown the existence of a quasiconformal mapping with complex dilata-tion p when p € B(C)l has a compact support. This conclusion is valid for ageneral p e B(C)r as well.

Theorern 4,3O. For eaery Beltmmi coefficient p € B(C)1, lhere exisls a home-

omorphism f "f e onb e which is a quasiconformal mappinC of C wilh compler

dilatation pr.Moreouer, f is uniquely determined by the following normalization condi-

tions:

/ (0) = 0, / (1) = 1, and / (oo) = oo.

We call this /, uniquely^determined by the normalization conditions, the

canonical p-qc mapping of C, or the canonical quasiconformal rnapping of C

with compler dilalalion p, and denote itby f u.

'C Jo ,tt Surddeu ab-r/ lecruoues eqt slstxa araql'gg'7 uraroaqa

.,(q acuag 'I(C)g Jo luauala ue se r/ pre8ar ue? a^r 'V - C uo 0

- r/ 3ur11estg'lrl - r/ 1as pue 'O + V: 3f Surddeur IeuroJuocrsenb e qcns xyg'!oo.t4

'O oluo y, lo tusyldlouoauroy D ol pepue?se s? CI uxDttrop uDprofp oluo V qslp pun ayl to 0utddou.t lou.t.totuoctsonb fri,ang .tt.7 uolllsodorg

'(69'p uraroaq;) ureroaqleruelsfxa aql Jo suorlecqdde lereles elels ar\ 'uorlcas slql Jo pue aql ?y

O 'auo pansap aq1 s f 'acua11 'suorlrpuoc uorlszrlerurou aql sagsrlesd'f1.ree13 (rl = 6il 1"qt os zrl peuufap eler1 ea\'g1'7 uorlrsodo.r4 3ur1ou fq'1oegu1) '('a'e) d - 6il leq? eas u?? a.ra pu" 'gl't tueroeql ul (H) fq leurro;uocrsenbsl ,r/ o ",tt = f 'ra,roarotr11 's?stxa osle "n/ 'lroddns

lreduroc e seq zrl s3u1g

{tz'v)

?as osle aM 'slslxe rrrJ ler{} u^roqs e^€q e^\ uaql '{srp lrun aql fl 7 alarl/I{

$zv)'v)z

v-c)z '(r)rt

las e^r (es€) slql uI '?uarogeor rrrr€rtlag lerauaS e u r/ 1eq1 asoddns 'dleurg'euo palrsep aql

sl / leql epnl)uoc e^t eoueH 'suorlrpuoc uorlszrlerurou eql seg$les / 'r(pea13

/z\ -7'C uo'e'€ (z)rt = (=llCa =Q){il

rl/ -

zzeeq

e.rrl snqJ 'elnr ursrl? Pnsn eql fldde uec a^r leqt os elqerlueregp ,t1e1o1 osle sr

?/r)rl -;=(z)l

Surddeur leuroJuocrs"nb aq1 'zf 11ur.od qcns fra,re 1V

'C uo'e'e elqerlueregrp,t11e1o1 sl il

'fV uorlrsodor4 ,tS '? go nrf Surddeut cb-rl l€?ruorr€c aql slsrxearaql 'aro;aq se acueg 'lroddns lceduroc e seq pue r(C)S o1 sSuolaq d uaql

{gz'v)'))z

+as eAr'r/l= (z),0 uorleuroJsrr€rl snlqgltr eql ,fq d laeq 3ur11nd,asecslqt uI 'ur3uo aql Jo pooqroqq3reu aruos ut ('"'r) O = r/ leql asoddns '1xag

'euo pensep e{f sl (I),r,4 /?)n,t teql salldur gZ'V l.uu'er-oaql ueql'r/ ro; uorlnlos lerurou aql aq a,iI pl 'asec srql u1 'lroddns lcedruoce seq r/ reql asoddns '1srrg 'at

Jo ecualsrxe eql .&oqs il"qs e^a 'acua11 'suorlrp-uoc uorlezrleurrou eql pue 8I'? uorlrsodo.r4 fq sArolloJ ssauenbrun eqa 'too.t4

1_(.'rl)"(#H)=",

',3\ = u*

'5G)d=Q)!

t0Isturddery l"ruroluo)rssn$ uo uraroaqJ arualsFg 'Z''

104 4. Quasiconformal MaPPings

Set g - f F o f -t . Then g is a quasiconformal mapping of D. By Proposition

4.13, we see that Fg = 0 a.e. on D. Hence, Corollary 2 to Theorem 4'4 implies thatg is a conformal mapping of D. Since /p(4) is a Jordan domain, Ca,rathdodory'slh"or"* gives the extension of g to a homeomorphism of D onto ;r'14;. Since

I = g-' o f t', we obtain the assertion.

Proposition 4.32. There exist no quasiconformal mappings of A onlo C.

Proof. Suppose that there exists such a quasiconformal mapping f : A -

C. Then,f-1 is also quasiconformal. We set p = pJ-t;then there exists the

canon i ca lp -qcmapp ing fP o f e . I f *use t g = fPo f , t henp , =0 a .e . on 4 ,

and hence g is a conformal mapping of A.

On the other hand, since g-1(C) = A, Liouvil le's theorem implies that g-I

should be a constant, a contradiction. D

Proposition 4.33. Let p be an arbitmry element of B(H)1. Then lhere exists

a quasiconfonnal mapping w of H onto H wilh compler dilatation pt.

Moroaer, such a mapping w (which can be edended to a homeomorphism

of H = I/ U R onto itself by Proposition 1.31) is uniquely delermined by lhe

f o I Io win g n o rrn ali z alio n co nditi o ns :

tr(0) = g, to(l) = 1, and ur(oo) - m.

We call this unique to satisfying the normalization conditions the canonical

p-qc mapping of H, and denote it by ut'.

Proof.The uniqueness follows by Proposition 4.18 and the normalization condi-

tions as before.

To show the existence. set

t ( z ) = { ; : ' 'I t(t,

z € H

z € R

z € H * - C - 8 .

By the uniqueness theorem, the canonical ;l-qc mappi.tg .ft of e satisfies

f i ( r )=74

In particular, we see that /a(R) = A. Since /P preserves orientation, f p(H) --

I/. Hence, the restriction of /i onto l/ is the desired one. tr

#,1'-##l{'"'�'*il*:s^rolloJ seuorler3elur;o ureuop eql apr,rrp eaa 'asodrnd s-rqt rod '?/t)al

/t = ?),r,tr eraq^,r

{z>l"ll r r 'npxpalr-,Gd)l I J

=ttlr

{1r1uenb aqt eterurtse o} secgns q 'I .- (I),r,4 pue (r lt)r,l /fi)n2, = (z)n./ acurg

(ez.r) .0 *- -llr/ll = -lldll se g * fipapoll - ,G)ltz>vtrfIJJ

teqt aor{s llerls a/y1 '0 Jo poorlroqqfteu aruos ur qsru€ d IIe teqf asoddng

'0 Jo pooqroqq3rau pexg auros uo

rlsrueA d U* I ,{1uo pue Jr papunoq fpr.ro;run a.re r/ 11e go slroddns eqt leqt e}oN'(gg'7 ura.roeql;o ;oo.rd aqf Jc) C lo il Eurddetu eb-r/ lecruouec eql u 7/ uaql

'(72/"21' ?l:.)rt - (4'!,t = (r)d pue (r/;.)nl /t = (,)a!

las 'lxeN

0 -- -llr/ll se O - Y'dllt -'Gt)ll

1eq1 sarldurr (ZZ'V) 'VZ'V ruaroeql o1 6 fre11o.ro3 ,tq

0 - -llr/ll $a I ts (1)ng ecurg 'fi)ng/?)a,t = (z),tt a^eq a^. '69'7 ura.roeq;,yo

;oord eql ur palets seit.r sy 'rl qcns fre,ra rog ,/d uollnlos leturou er{l slsrxa araql'-llr/ll

lpus {lluarcgns q}l^a r/ fre,ra roJ I > ,C. -llrlll acurg 'fpre.r1rq.re (6 a)d xrg 'papunoq ,(prro;run er€ r/ 1e ;o sl.roddns eql 1"q1 etunss? '1srrg

/oo.l2'

0,,(oo,o o,u: I I) = v'dttqtl

puor(C)g3danym

'0 * -llt/ll 80 0 - Y'dllt - "Gl)ll

'(Z <) d finaa tog 'tg'7 BurruaT

'eurual

3ur,rao11o; aq1 errord ?srg e^n 'n1l' lecruouec eql roJ 'lZ'' ureroeql ol 6 ifre11o.ro3

ur r/ o1 lcadsa.r {lln ;^{ suol}nlos Isurou aq1 ;o ,(lmulluoc u^{oqs fpeaqe a,reqai!\ 'rl

luerf,Ueoc nu"rlleg aql uo ;3f ;o acuapuadap ure?uo? q lo il Eurddeur

I?ruroJuof,rspnb lecruouec aql uo st?eJ InJesn pue luelrodul tsotu eqt Jo aluos

sluerJsaoc ilrrBJlleg uo eruapuedeo '8'?

sluarf,SaoC ur"rtleg uo acuapuadaq 'g'7

.21 t r7-

tu>l"l>z/rir r'llll=(,t)r nu )

90t

106 4. Quasiconformal MaPPings

Notethat,for asufficientlylargeR,everyFts isholomorphic on{z € C I lzl > R}and Fp converges to Fo(z) = z uniformly on {z e C I lzl ) i?}. Hence, we cansee that Iz - 0 * llpll- * 0. On the other hand, since

. f I l z211ru1"1z) - 1 ) , z2 , lP dxdytt =

J Jt'tr.t,t."t l-lFu(')r i

@t'(z)Y 'l

lu '

Corollary 2 to Theorem4.24 shows that 1r - 0 * llpll- - 0. Thus we obtain(4.28).

Finally, for a general p, set

u ( z \ = [ u Q ) ' z € A

1 0 , z € C - 4 .

Letting /v be the canonical z-qc mapping of C, we set gF = f P o(f')-r' Again,by Corollary 2 to Theorern 4.24, tv '--+ id uniformly on C as llpll- --- 0. Hencewe may assume that every f'(A) is contained in {z € C I lrl < 2} and contains

{z e C I lzl < t/Z}. Then by Proposition 4.13, every per vanishes on some fixedneighborhood of 0. Further, since

(f p)" = (gF)" o f , . ( f , )" * (c\ , o f ' . (V), ,

and since either (/'); = 0 or (gp)t" f' =0 a.e. on C, we have

l l(f ')" - 1llp,a S l l((su)" - 1) " f ' '(f ')"l lp,a + l l(f '), - 1llp,a.

Express the right hand side of this inequality as /3*/a. We have already shown

that .Ia * 0 as llpll- --* 0. As for -I3, we obtain

" l l l( t " )o s=: - I I l (su)" - \P l ( f ' ) , " ( f ' ) -L lP-2dxdv

L _ t c " J J y " 1 a y

t ( r t ^ IxdY ' l l ^ l t r l " l ' o - 'd 'dv |. - t l l l ( s r ) , - l l 2 p d . x d y . I l l ( f ' ) " | ' o - ' ' ] ' r '

- | _ kz tJJt t , t . r t ' . " , -

with ft = l lpll-. Hence, by using (4.28), where we replace iF and pby gts and2p, respectively, we can show that .I3 * 0 as llpll- * 0. Thus we have the

assertion. D

To investigate dependence of. fp on;r, first we shall derive the following

integral formula for f P.

Lemrna 4.35. Fix p with p > 8 arbitrari ly. Let p be an elemenl of B(C)1

satisfuing l lpll- . Co 1 I. Then the canonical p-qc mappins Ip of C satisfies the

following integral formula:

l)z - rl 112) a ll(gt)o=WT

{c=l"ll r

I1eq1 saqdrur (gg'p) 'arag

()z-1)"((z)a{)'vf f _ _), -t@)n!'onI 'nnxn::----)?)-#: Jl ,, zp z I

leql 6I'? uorlrsodo.r4goyoo.rd eql ur se ees u€f, e,lr ,e1nurro; s(uearC Surfldde ,(g'{t > l"l > g I C ) z} = ey tes

(9 enrysod geurs flluarcgns fra,re .ro;'irro11'leus dlluarcgns sr -llr/ll *

3uo1 se r/;o luapuadepur $ u, t"ql alou ,.raq1.rng .(Og'l) sa,rr3 qrrqal ,ur3r.ro aq1

Jo pooqroqq3rau auros ur snonurluoc raplog-(d/Z- il.1 r_(,r1) leq+ epnlcuo?e^r snql 'ge't uraroeql;oSoord eql ur u^toqs s€^ s? ,ur3rro eqtJo pooqJoqq3reueruos ur snonurluof, rep19g-(d/6 - 1),tpr.royrun sr ,_(21)

,prnq Jaqlo eq? uO'utErro aql Jo pooqroqqSrau aruos ur snonurluoc zlrqrsdrl fpr.royun a?ueq pue'leur.roguoo .l

r_(V) l€ql ^rou{ e,lr ueql .ur8rro aq? Jo pooq.roqq8rau a(uos ursaqsruel t{rl pue poddns laeduroc e serT c{rl ,f1aue11 .gg.t ureroeql go;oo.rdeql ul se fl.repurrs peugep aw lt eraqa.r,'rl ozt s (z)7/ esodurocap ,1ae; u1

(oe'r)'A> z' k_qtdlzltu Zl(r)nll

/pooqroqqsreuers,xeareqlffilX'&'.T":':11..i':.1"T:,J,;r.""ui;1r1ir"T::lou se^eqeq (r)n! /r leql {raq? plnoqs ar* 's1q1 op oI 'epturoJ s(ueer9 fldde aar'7 uo 1e.r3a1ur ?er€ ue fq aprs pueq lq3rr aql uo ler3alur aq1 acelde.r o1 ,,rno11

'sluelsuo? ar€ €r pu€ y'eraH

.)z -tG)a! on[ )*)s +v= zP z I"

,o (), - I * r. z\ /z\ vQr Q, - t), onf

_ _^) -, onf.\ ,") "rl)(.tJ"/ l=;ffi l=,rb),r J

e^eq e^r acuag 'Q)nI /, o1 pnbe * n/(n)nl uaqa.z f 1- m

ol z elq"rr"^ aq1 a3ueqc 'asodrnd qq1 .16g 'V - C uo 1e.r3a1ur eare elq€trns e dq

aprs pu"q 1q3r.r aq1 uo 1e.r3a1ur tsrg eql acelda.r plnoqs en ,uorlresse aq1 anord o;

v)) ,op,pffi"ff +-;e"'l o+=Q)a!serrr3 (61'p uorlrsodo.r6) "lnuroJ s,nreduro4' lsug'too.t 4

'Q/t)nt /t = Q)r! a.raqm ,V ) ) fr^taaa nt

op,p (' --r - !;) u(, \'=l'!)" [[ "--

. \ ,) ,") / (r)'G!) il rGz'v) - ( z r-z )-z\,.-. -.vffy nn,e\r_1+; -,

)(,)'Gt) I I t-)=Q)atr0Isluerf,saoC ru"rllag uo acuapuadag 'g'p

108 4. Quasiconformal MaPPings

as 6 - 0. Since p > 8, we can show by Htilder's inequality and (4.30) that the

area integral on the right hand side converges absolutely as 6 -* 0. Hence we

have

rp(( \=o!?c - ! t t U') ,Q)0,0,,-c ' [ [ Gu),( ' ) --L,d,ayr \ s / - 2 " i r J J a z - \ t - ; l l a j t ' @ y l - r c

for every C g A.Moreover, since both sides of the above equality, considered

as functions of (, are continuous on ^4, the equality still holds for every C e A.

Hence, by using the normalization conditions that .f'(0) = 0 and /p(1) = 1, we

obtain the desired formula.

Using integral formula (4.29), we prove the following:

Proposition 4.36. It 1t conuerges Io 0 in B(C)1, then the canonical p-qc nxap'

ping fP conaerges to the identity mapping locally anifonnlg on C.

Proof. F:xp > 8. Since llpll- * 0, we may assume that, for every p considered,

l lpll- .Cp 11, and hence (-4.29) in Lemma 4.35 is valid. Writing the right hand

ria" or 14.20; as ( + r11; + i(C), we shall show that both 1(() ana i(1; converseto 0.

First, since( f p ) r = p ( ( f p ) " _ 1 ) + p ,

we can find, by using H<ilder's inequality, a constant M such that

l / (Ol S M( l l ( f r ) " - l l lp ,a+ l ) l lp l l - , e ea

for every p. Hence by Lemma 4.34, we see that .I(() converges to 0 uniformly on

4 as llpll* -* 0.Next, recall that V and rn in (a.30) can be chosen uniformly with respect to

p. Thus we can find a constant M such that

l i ( ( ) l < M$G\ " - l l l p ,a+ l ) l l i l l - , e ea

for every;r, where ! is defined by(a.25) in the proof of Theorem 4.30. Since iris the canonical p-qc mapping of C, and since llpll- = lllll-' we see by Lemma4.34 that i(C) - 0 uniformly on .4 as llpll- * O.

Hence, we have proved lhat f rt - id uniformly on 4 as llpll- -'- 0.Finally, for every positive r (< 1), set p',(z)- tt|/r) (z eC), and consider

the canonical ,rr-qc mapping fl" of C. Set

f ,(r) = fP'(rz) l fP"(r) .

Since p1.(z) - F,Qz) = p(z) (a.e.), and since /" satisfies the normalizationconditions, the uniqueness theorem gives /" = fP.On the other hand, since

fp" - fd uniformly on ^4 as shown before, so does f, on {z e C I lzl < Ilr}.Since r is arbitrary, we conclude thal f tt ---+ fd locally uniformly on C. tr

o1 saSrerruoo ?/ Q)rl |€rll ees uef, ara'V ) ) frarra.ro; y uo alqerEalurdlalnlosqe s1 lue.rofeu sql ecurs 'yg a?w1r(lluaragns elqelrns " qll,$ 3 fra,ra ro3

,l,-rl(l)l -r)v>)'ffa-qtdz-lzlw

lueroleur e seq 1e.l3a1ur aloqs aq] go puer3alulaql teql sarldurr (gg'y) aleurrlsa uroJrun eq1 'raql.l\{ '0 *- t s? y uo ,t1u.ro;run

t (r-t _ )r-r\ z(@)a,r!)"f f "__\ _ | g+- \fipxpr \z) ,r)) Q)t ttrJ O)'^/

;oo.rd eqr u-r reql ol rerrurs luaurnEre "" fi1:Tfi:{Lt-?ffii:jl,il'Qz/rz1'Q/i"=Q)c

Pu"

Qz / "21' Q /i!),t = ()'!il = G)(t)d1as '1xap

'0-ts"f,Zuor(luroyun

(se'r)**(*+T-.) @:[f +-o1 seEreruoc t/Q)\ leqf gg'? uorlrsodor4 go;oo.rd eql ur leql

ol relrrms luaurnSre ue,tq,noqs usc e,r\'0 - I se 0 + -ll, - l/(l)rtll e?urs pu€

(t)a + $ - "QsrlD(r)rt ='(<r>nl)erurs 'lsJrJ

'O)'^r+())'r+)"'(3)r/ rol (66'p) elnutoJJo aprs pueq 1qEr.r eql ssa,rdxg'1frara.rol (3)r/ - r/ qq^p1e^ q (OZ'f) t"qt erunss? feur aar 'gg'7 uorlrsodor4 ;o ;oord eq1 u sy 'too.r4

et.v) .c r ) ,oo,oWUy"[;- =e)t^l!

:uotlvlu?seJdat, pliayut aqy soq lnlt'.teaoato141 '3 uo tu.tottun Qpcol * acue6.taauo? e1l puo 'C)

) fi.r,aaa "tot sTnia

(re'r) =--*-----. t'ii = (:)t"lf) - O)r,l,r/

'' ,ueqJ

'0 *- r s? 0 ts -ll(l)rll rour u?ns (c)-z ) (7)t puo (c)-z ) n ?lqopnc qnn

c ) z'(z)(1pr* (z)'ty = (z)(7)il

nt^tot aq1 u, u?I1r.l,r.l" sl Q)il ''a'l'0- tID elqv?tua.ta$ry q G)rt

tayt puo'0 *- l sD 0 {- -ll(l)rlll 7oq7 asoililng 'TnTaunroil aalilu.roc o ro IDaro uo |utpuadap sTuarc$aoc runuH?g lo fr,1gutol o aq {(7)d} pI .Le.? ruorooq;,

60tsluarf,Ig:eoC rurcrllag uo acuapuadaq 'g'p

110 4. Quasiconformal Mappings

_ ! t f . . / 1 \ r ( c , _ / , \" JJ^"1;) t (r= -fr1a'ao G34)

locally uniformly on 4 as I * 0.Thus, changing the variable z to If z in (a.3a) and adding it to (4.33), we

have (4.32) for every C € A.Finally, the same argument as in the last part of the proof of Proposition

4.36 shows that (/p(t)(()-./)/t convergesto the right hand side of (4.32) locallyu n i f o r m l y o n C a s t * 0 .

Corollary. Let {p(t)} be a family of Beltrami coefficienls depending on a realor a compler parametert . Suppose lhat p(t) is di f ferent iable att =0, i .e. , p(t)is written in the form

p Q ) Q ) = p ( z ) - t t v ( z ) + t e ( t ) ( z ) , z € C

wi th su i tab le p€ B(C)1 , v e L* (C) , and e( t ) e .L - (C) such tha t l le ( t ) l l - - 0as t ---+ 0. Then

1t,(r)11; = Iu(0 +t j t ' lu l (O+ o|r l ) , ( € c

locally unifor"rnly on C as t * 0, where

i'vc)=-+|1",v,ffi0*0

Proof. Set 1, -

7u(t) o(.fr)-t.Then the complex dilatation ,\(t) of f l is given by

' \ ( ' ) / t ' ( t ) - t '%) ' , ru , - ' '=t'r,= \ i :Fmff i)u\r

/

Hence, .\(t) is written as

. \ ( r ) = r i + o ( l r l ) a s r * 0

i n .L - (C ) , whe re

\ - ( u ( f P ) " \^=( , - tP ( * : )o ( ru ) - 'Apply Theorem 4.37 to this family {/r}. Then we conclude that (f i(() - C)/tconverges to

/ t i l ( ( ) =-1 [ [ ^ ta- ( ( ( -1)7t J Jc z\ ; 1)k- C)axav

locally uniformly on C as t * 0. Hence, changing the variable z in this integralto (f r ')-r(z) and noting that (/r(t) - Ir)/t = {(/, - fo)/t} o f u, we have theassertion.

(qe'r))-r

u I ) 'rpG)a

frq pau{ap dt to dtp uotTotu.totsuvr? peql?H aqt '(U)J,? ) d't fitaaa.rol (g)puD 'I - zV puo 'd o7 Tcadsat Vpn snonutluoa st. dy (r)

?Dtll lons dy TuoTsuoc o s, ^reql'(2 1) d frtaaa .tog (zsar11) .gg.7 BrrruraT

'u1(ou{

11eirr ,tlecrsselc ere qcrrl/rr 'uorleuro;suerl lJaqlrH eql Jo sarlJado.rd ssncsrp lsrg

[eqs a^r eroJalaqt '(66'7 uorlrsodor4) ureroeql s,punur3,{Z-ugJepleC a.rrord o;'uorleuroJsusrl

?reqlrHIerlss€1f, eql sl J srrll of Surpuodsar.roc rolerado aq1 'esec

leuorsueurp-euo eql uI'atl = g uo pr3alur reln3urs e s€ paugap ser T,?,'V uorlrsodo.r4 ur 7 .role.redo eqa

tuoroaql punrutfz-ugrapl€C Jo Joord 'p.?

'o'"all .ll ur.rou eql ol lradsar qll^{.{ll€rlr{d.rouroloq ro f11ecr1.{1eue-leer ? uo spuadap (r),// ueql ',t1arlr1cadsa.r 'fllecrqdrourolor{ ro f1ecr1,{1eue-leer ? .relarue.red xelduroc € ro l€er e uo spuedap (l)rl uaqin 'elourraqtrnd

"''"llf..tp1 , r llo*, 'o -

llL(o)Dl (torn.f) - - -ll urr

uaql '? ralaureredxalduoc e ro I€ar e o1 lcadsar qll/tr 0 = ? le elqerlueraJrp st (l)r/ uaq,rl '1xe11

'0 *- t se C uo f1u.ro;runo1 sa3reauoc ,73f '.re1ncr1red u1 'oo + u sB 0 * n'"slln! - "nlll<- u s€ C uo'e'e d <- url teql pue u.r(.ra.rre roJ { > -ll"r/ll

s?uerrlgaoo rurerlleg 3o acuanbas e eq IT{"d} tel'reroarotr41

e q {{ t -ll/ll | '''slln/ll} uaqyr > oCrt pq1 asoddn, p,* 'Tti"i"l';

tlll^{ { xlJ 'Z 4 d pue ) Jo I Surddeur IsuroJuorrsenb lecruouec ,tra,re roy

/ a 5l"lr r\ / all"lr r\

lop,pol,ll lll* lop,pol'{l lll* a/r\ rr / a/r\ rr /

,q/ r. r_1", ; -::),uire"f

r "r =,,"s1 l/ll l(zz)I - P)tl

las 'ar

luslsuoc erlrlrsod e xrg :([91] sregpue sroJlqy ';c) s11nse.r Eurmollo; aq1 e.rrr3 suorle3rlse,rur radaap aurcS 'qJDuaA

'C f ) paxg fra,re roJ t ol lcadsar qlr,u crqd.rouroloq fl Q)fr>nt uaql 'l ralaureredxalduroc eql o? lcedsar qt-ra aleqa{Ja^e elqerlueraJlp q (l)r/ ''a'r 'fllerrqd.rour-olotl ? ralaurered xalduroc e uo spuedap (irt uaqm. '1eq1 palou eq ol sl 1I

['<ll-,ll f y o*,

/ t*r= (?)dn

,t11eco1 ;rfueql 'oo

leql qcns

IIIuraroeqJ punurt.{2-ugrepFC Jo Joord 't'?

ll2 4. Quasiconformal MaPPings

salisfiesl l / /pllp,n < Apllpllp,n,

wheru l l . l lp,n means the LP'norm on R'.

prool. Since the assertion for the complex-valued functions follows from that for

real ones, we may a.ssume that I is real-valued.

First. set

r(() = u(C)+ iu(() =+ I :*0,,

( =€* i t t ,n> 0.

Since u(() is the Poisson integral of rp, it is harmonic on the upper half-pla^ne

.F/, continuous on I/ U R, and coincident with rp ott A. Th" real part u(() is also

harmonic on ,E[. Moreover, since

,(o=* l:d#e@)do= ! f P G + , ) - ' P G - o ) ̂" = d ,

r J o t x ' + n o

for every ( = €+a4 with ? > 0, we see that u((*f4) * neG) as ?--* 0 for

every ( € R.Next, set

w(0=lr(c)lo -f,t"t}Y.

Then we obtain by simple computation that

azw ,*, -2

f f i ,G) = f , t l r tc) lo-2- l " ( ( ) lP-2) lF ' (c)12 >0, ceH.

Hence, applying Green's formula on the domain

D e , R = { ( e f f l n > r , l c l < f t } ' 0 < e < E ( o o ,

wehave f ( i \ a l y

J,o. *(j, -uc o< >o'

Now it is easy to get a rough estimate

l a w | , , , , _ 1 t

lal = o(cl-') (l(l- *).Hence, letting .R + oor we have

t Y u s oJ 1r1=ey otl

Note also that another easily obtained rough estimate

'eturuel Surirrolo; aq1 sarrr3gg'7 eururerl'g.ro1e.redo qql ol lcadsa.r qll1\'(C).35) 3 a1 pue c I ) araq^\

tztz {t<t,u S yIq g;i = 'ffiQ+'14 Jt I "

/ , {;4lcllg3ar^ \

opey-,(*O;;ooe 'l +'*'i) ""1 |=,t,^t

aqr ,.a..r ,ro1erado ,,a3?*e lerncrr),, "ur r"oirT:i"*t:f?lHJ,[Y"1.::*t'T:[iaql o1 U uo uorlsuroJsrrerl 1reqllH erllJo suorlezrl€raue3 aq1 Jo euo sV'C uouoll)unJ e se dtg p.re3ar a,n'elqerrea xalduroc e s€ l Surraprsuo3'(qg'l) "t

? -, = q oI e alqerr"^ aq1 a3ueqc pu" (C)JC ur d uorlcun; e aqel'mop

'd ,{.ra^rra JoJ lu€}suoc peJrsep e sr

( - /r-a\\

"to\'- ot"\T)) -o-,

'acua11

-,,('-,r,(#)) - -, .a'a114511zlax -!" t tu'dlldflll

sa,rr3 eururel s(noleJ '0 ol puel r 1e1 ',{1eutg

'?p ar(,p + il^r*f "''(t -'t'(#))

t tu dre? + ?),r*[

l€q} aPnlruoc e^\ snql

",,{,,"(rrot('r + tq

* /) *

","(*ur'p + ))nt :/) }

fq papunoq $ aprs pueq lJel eql leql aas er\ 'flqenbeur s(r{srtro{ur4 fq 'a.ra11

'JpalQt+ t)nl *-[

+ ltudllp+ l).rl -7

oJt-J

?€ql apnlcuoc aar'(oo'r] uo lr o1 lcadsar qlrm Surler3alur'acua11 'p.r3e1ur 1se1aql uI uollsllueres1p pu€ uor1e.r3a1ur Jo repro eq1 e3ueqc uec e/r pq1 sa11dur1qcrqal, 'lr o1 lcedsar ql1,u ,(1turo;run ,t1eco1 sploq '.Ztl e3.re1 dlluarcgns e qlr^r

'u > l',-(I + "l?l)wt lfr, . r#rl

TIIuraroaqJ punurtfT-ugrappC Jo Joord '?'t

ll4 4. Quasiconformal Mappings

Lemrna 4.39. For a giaen p (> 2), lhe inequality

llsell, < ioollolloholds for eaery I € Cf (C).

In particular, lhe operator S is extended lo a bounded, l inear operator onLP (C) into ilself.

Proof. For every 9, set gs(z) - 9Qei0). Then

l lsello. * 'uo llneellp.z |e lo, r l

On the other hand, for every d, Lemma 4.38 gives

lneell!, = [* ( [* Wrrloa,\ av'

J - o \ J - o /

f @ / 1 6 \

< A l . I l l l e s l P d x l d y- F J - - \ . / - -

" /

=allleelll = elllvlll.

Hence, the assertion follows. 0

The first assertion of Calder6n-Zygmund's theorem is reduced to the followinglemma.

Lernrna 4.4O. For eaery I € Cfl(C),

Tp = -S(Sp).

Prool. First, by Green's formula we obtain

' t t t a / - 2 \sp(0 = lg n JJrr,r,r,Pe + (); \a )o,oot (= n ^ 1 1 [ [ e " e * c l l a , a y + [ l P Q , \ ) o r \

.*o ur ["/"/11 "l>,J " '

lzl -

J 9"p,1 2 lzl )I t t 1 | t t 1=;

J J.e"(z + C) ^a'av = ; I J.peQ * C) ̂dxdu

=L? ( [ [ e? ) . d "d , ) .r 0 . , \ J J c l r - C l " /

As in the case of the operator P, we replace the kernel Lll, - (l of the integral

on the right hand side by

1 lk( t ,C) =

l , - . . l - A ' z , ( e c '

.(oo - u) ry-- l*-a- tl of @ - zLln - z) g - =)LoPu"taI

(E;Y'n'uu [ [) + = (W *''^-n',,1 *lsql alq"lr"^;o a3ueqc fq aas a,u, 'urJa1

lsrg eql roJ sY

I l> -,tt>l {a>ttt}S I ze _ln -.)ll) - zl {u;t*-ttt

S S's ) --a'

t-rt;p ll Q--tplp tt ni-ufq aprs pueq tqSy aq1 aceldag

{**F*{a'n-)| [?] *r = (,,wffi"[ [) +1eB eaa 'acue11 'oo {-

U s€ C uo f,prroyun d1eco1 g o1 se3ra,ruoc

/ 11_ 21 {a<tn_tnss\ 1g-f"etefu)t u / e

tgq? eas uec arr,r tlsrtg

**m"lI1er3e1ur eq1 yo f111q€Iluareglp prl.red eql {caq? o} Peau aal 'ale11

(ze r) {oo,o (**F#"1[) 3 ,4^"[f] ++- = (rn)(ars)gurelqo aa,r 'e1nullo; s(ueerc Sursn ,(g 'c ) n dra,ra .ro;

{0,,0 (**F#"[ [) u," ^" | [ ] ry + ={uo* (E#u," ^"t t) ('')h "f

[] ry + =( )f f\ morL

\uotr@')ho)as I I I it=('xas)s1eql ueroeqt s(lulqnd Sursn ,(q eas ar\ 'ecue11 '(C)JC > a

d.ra,re ro; (C)al o1 s3uoleq dS tn,{t saqdur 68't €IuureT 'ptreq raqlo eq} uO

rrerr,a ro; pte^ IIII' q (ge't) ,"ur,Xll,t'lJ:l{;":ffi:Jffitffi? ;?l{"1'1I al frarra .ro; 1e.rEa1ur aq1 ;o ecuaEtreluoc et{1 aalu€r"n3 o1 sr uorlecgrpour srqa)

' (owor|z)q(z)dt"il) ++= o)ds a^eq a^a uaqr

(ge'r)

9IIuaroaqtr punurtd2-uoroppC Io 1oord '?'t

116 4. Quasiconformal Mappings

Similarly, we obtain

! ( t t : {4: ) - -o 1p*m).0z \J Juetsny l.,l l, - Cl ) z \'- --/ '

Hence. we conclude that

*(ll"f*"*)exists and is equal to

z - r = - '

Now, going back to (4.37), we have shown that

s(s,p)(,o) =* {+ | l.ora (* - i) **\a ^=-i ; , rr(w)=-Te@).

Thus the assertion follows. tr

Finally, the last assertion of Proposition 4.22 follows from the continuity of

Co with respect top. Noting that Co ) 1, which is verified easily, the continuity,in turn, follows from the following Riesz-Thorin's convexity theorem.

Lemma 4.4I. The funclionlogC, is conuet wilh respecl lo l lp on (0,I12).

Proof. Fix p1 and p2 with pi ) 2 for each j. Set oi = llpi and Ci = Cpi(j : 1,2). It suffices to show that

l lrfl lu" S C,'-' .cl.l lf l lq.

ho lds fo revery a= (1 - r )o r+ ta2(0 < t S 1) andevery f e t ' r1 "1C1. S ince

I t l

llrfllu' s6Ltt(t-a1rllirn,r.,-., rrlJ J"Tf

' sdxdv

by duality, we shall estimate il"f f .gdxdy.First, we assume that / and g are step functions with compact supports. For

every complex value (, we set

r(O = 177GY' f,t / l

andG(O = ;r1(t-a(e))/(t-d

el c l '

where o(O = (1 - C)ar + Caz. Clearly, ,F' (O and G(() are also step functions forevery (. With suitable real constants .11 , we can write

'y xtPuaddY;o

pue'g raldeq3Jo setoN aes'os1y'[916] ueartlns pue'hOI] eme3eg'[691] sauol'[gg] Surrqag '[99] ut.*tq 'lttl'ltZl s.rag '[61] sroJIqY '[gO-V] raqoqts '[tO-V]

r€{eN pue orreg '[16-y] rauaqc,(g pue uuerulag '[Sg-V] zfztypve zcrr*oufrinel'[qg-V] Surrqag '[gt-V] ralqceqos pu€ uqof 'srag ees'aldurexa roJ 'suollnqlrlslp

enl€A Jo pue 'suorlcunJ luel"Alun ;o 'sdnor3 u€Iulely ;o

'sace;rns uueruelg Josalroaql aql s€ qtns elqelJe^ auo Jo srsfleue xaldtuoc Jo splag snolrel uI osle lnq'saceds ra[nuq)IeJ;o ,t.loaq1 aq] ur fluo 1ou paqdde pue '1oo1

l€]uau€punJ pue'luelrodurt '1n;asn e se paztuSocar 's{eperrrou 'are s3urddeu 1eu.lo;uocrsen$

'[gSa].H.\t Pu€'[69I] !\otsotrl'[g] pre3y ol uolluel]e ll\€rP osle

e/11 '[60I-v] gpslg1 'actrelsut ro; 'aas 'sSutddeur leuroJuoJrsenb prleds rog

'[Og-V] uauelrt1 Pue otqarl se qtns s3utddeur leturoJuo?Isenb uo s1xa1

prepuels aas 'sar1.rado.rd ctseq Jerl?o pue suolllugep asaql roJ 'pasn flluanbaq

osl€ are 'q13ue1 l€ruallxe eql ,(q euo Pue (7$ '1 .ra1deq3 jc) uorlel"llP lelncrl) eql

fq auo 's3urddeur leuroJuocrs"nb;o suolllugep luale^Inbe reqlo 1o aurog '[6-y]

alou ern??al paterqelac (sroJlI{Y Jo A Pue 11 s.ra1deq3 uo paseq st laldeqc stqa

saloN

D 'uollJass" aql sa^IraP luaun3re uolleunxo.rdde

eurlnor e 'slroddns lceduoa qt!^{ suol}?un; dals frerltq.re ere f pue / acutg'?=)W

."/'ll/ll .\c.,_Ic I l(t)ol

,D/rlUll?olzpl+tp(l-r)

l€ql aPnlruot II€? era

- c7?o1j - r28"r (l - t) - lO)olsot

uorlcunJ truowreqqns aql ol ,fllcallP

eldrcurld leurrx"ru aq1 turrtldde fq ro 'uraroaql sauq-?arql l€elss?lc eq] /tq 'acua11

', r,C I tlltll)zc > l())ol

ulelqo "^'{I = } | C ) )} uo'{gePu15

.,1,,( ttlvll),c > t'"-r)/rll())cll'"/'llo)a-rll t l())ol

aeqaa{'{0=llC>)}uO'{I > ) > O I C > )} uo crqdrouoloq pu€ papunoq q ())O'relnrrlred u1

'fiptp0.la

l"qt snol,rqo sI lI '(oo >) 7,9 ,(ra,ra ro3

{W > l}ll C > bp + I = )} uo arqdrouroloq Prre PePunoq tI O)O '{11uenbasuo3

"f | = r,t,

'(tuns alrus e) 2,rara! = ftpap())g. O).r.2 "[l

= OW

LTIseloN

118 4. Quasiconformal Mappings

Recently, quasiconformal mappings have played a crucial role in new inves-tigations in the complex dynamics. A new notable result, called the improaed\-lemma, has been proved, the statement of which, for the sake of convenience,we include here.

The improved )-lemma. Let E be a subset of C, and tet f (\,2): Ax E - ebe admissible, i.e., let f satisfy the following condil ions:

( i ) / (0 , z) = z , z € E,(i i) for euery fired \ e A, the map /(), .) : E - A it on injection, and

( i i i ) /or euery f i red z e E, the map f ( . , r ) , A - e is holomorphic.

Then there is an arlmissible map iQ,4 i Artsx e -----* A such that f -

i onA t t e x E , w h e r e , A t t s = { } € C l l l l < 1 / 3 } .

Moreoaer, for eaery fixed \ € Atls,l(f , .) is a quasiconforrnal homeomor-phism of A oilo itself.

The second assertion is the contents of the so.called ),-lemma. For the proof,see Bers and Royden [43], and Sullivan and Thurstonl2lTl. See also Slodkowski

[20e].As for related papers in this field, we further cite Blanchard [44], Douady

[52], Douady and Hubbard [55], Mafr6, Sad and Sullivan [135], Shishikura [206]and [207], and Sullivan l2l4l,l2I5l, and [216].

'asodrnd qql 1U f11eeu s3urddeur IsruroJuocrs€n$ 'ace;lns aseq

eql uo sernlcnrls xelduroc aqt Jo 1as a{l se aceds reilnuq?ral eq} reprsuoc e^rueqin 'sa.rn1cn.r1s xelduroc eqt Jo suorlJol$p eq? ol uollrpuo? sseupepunoq ruroJ-Iun aql esodurr ol sI qqt op o1 ,teu aug 'saceds rellnurqcral eql Jo uorlrugep eq?ur 'sauo pcrSolodol rrcql rer{?sr 'suorlrpuoc c1!t1eue pu€ e^r}f,rJ}seJ eJoru reprs-uoc plnoqs em 'q3noua

InJtln{ suorleErlsarr,u! eq} e{pru o1 'acuanbesuoc s sV'C uo lou op lnq

'y uo lsrxa suorlcunJ crqdrouoloqpepunoq luelsuocuou pue suorl?unJ uearC s? qcns suor??unJ crseq 'f11en1cv 'sarl

-.rado.rd ctytleue xalduroc luaregrp snorJ?A e^€q lnq 'crqdrouoagrp f11en1mu arey {slp lrun eql pue C aueld xalduo? aql 'acuelsur .rog 'srs,tleue xelduoc 3olurodalar,r aql urorJ ts?al 1" 'fpureg a13urs e ur se?sJrns crqdrouroegrp flpnlnur

1e eceld o1 q3no.r oot sl 11'secegrns uu"ruerll praue3Jo esec aq1 ur'ralemog'(Iraldeq3 yo g$ ees) sEurq.reur se sdno.r3 l"luauspunJ eql ueealeq sursrqdlourosrqlgm paddrnbe g, o1 crqdrouroagrp sac"Jrns uueruarg (paryeur) il€ Jo las aql 'g.

;o eceds rellnurq?ral aql se 'pa.raprsuoc eler{ eru'9, ace;rns uu€ruerg pasolc e rod

sacedg rallnuqrlel Jo uorlrnrlsuo3 cry(leuv 'I'g

's8utddeur railnuq?IeJ pa11ec'sSurddeur

I€ruJoJuo?rsenb purerlxa eql Jo sseuenbrun pue ecuelsrxe eq1 sr;oord eqt ;o ,ta4eq; 'aceds ueapqcng l"uorsuaurp-(g - 0g) I€ar eql q II€q lrun uado eq1 o1 crqd-Joruoaruoq q (a ?) f snuaS;o ec€Jrns uu€tuerg pasop e go eceds ralpuqcreJ eql

}€q} sel€ls qcrqin 'ruaroeql s(rallnuqcrel arrord plr€ '(Z () 6' snua3 Jo sat€Jrnsuusruerl{ pesolc Jo esec aq1 ele3rlselur ear 'g pue U suor}ces u1 's3utddeu

leur-.ro;uoarsenb Eursn fq a?eJrns uuerueql frerlrq.re ue ;o aceds Jallntuqrle1, eql Jouorlruuep areu e arrrS aal '1 uorlaaS ur '1srrg 's3urddeur

l€ruJoJuocls€nb Eutsnfq f1e,r�1eure11e saceds Jallnr.uqcral lcnrlsuoc II€rIs eiIr 'reldeqc $ql uI

saJBds Ja[[nuqJ.raI

g ra+dBqc

120 5. Teichmiller Spaces

5.1.1. Teichmiiller Space of an Arbitrary Riemann Surface

Fix an arbitrary, not necessarily closed, Riemann surface .R. For every quasicon-formal mapping f of R onto another Riemann surface.9, consider a pair (^9,/).We say that two pairs (S1, fi) and (Sz, fz) are equiaalentif f2o f ,

I is homotopicto a conformal mapping of 51 onto,S2. Denote bV [S,/] the equivalence class of(s, t).

We call the set of all such equivalence classes lhe Teichmiiller space of R,and denote it by "(r?). This "(,R) can be identified with the secalled reducedTeichmiiller space ?#(I) of a Fuchsian model I of R, as shall be seen in $1.2.Letting idbe the identity mappingof .R, we call [.R,id] the base point of "(A).The topology of "(.R) shall be introduced in $1.3 by means of the Teichmiillerdistance.

In this definition, we have used the notion of quasiconformal mappingsbetween Riemann surfaces. Since quasiconformality is a local property and con-

formally invariant, we can naturally define quasiconformal mappings between

Riemann surfaces. However, there is another way: namely, by the uniformiza-tion theorem in $1 of Chapter 2, we define them from quasiconformal mappingsbetween plana,r domains as follows.

For every Riemann surface E, take a universal-covering surface ,E of ,R. Bythe uniformization theorem, we may assume that I is one of C, C, or the upperhalf-plane I/. For every homeomorphism f of R onto another Riemann surface

.9, by Theor em2.4,there is a homeomorphism /, a lift of /, oj E onto a universal

covering surface .9 of 5 (which is also assumed to be one of C, C, or Il). We say

that / is qaasiconformalor K-qcif the lift / is quasiconformalor /(-qc. (Here a

quasiconformal mapping of d means a canonical quasiconformal mapping of ecomposed with a M<ibius transformation.) Note that by conformal invariance ofquaslconformal mappings, this definition is independent of the choice of a lift f.

First, we discuss briefly the Teichmiiller spaces of the exceptional Riemann

surfaces (cf. $4.1 of Chapter 2).

Example.l. Suppose that E = e. Then fr.= R = 0, and the only Riemannsurface homeomorphic to .R is^e . Moreover, every quasiconformal mapping of Cis homotopic to id. Hence ?(C) consists of a single point.

Example 9. Suppose that E - C. By Theorem 2.I3, Ris conformally equivalent

to one of C, C - {0}, or tori. The image of C or C - {0} by a quasiconformal

mapping is conformally equivalent to C or C - {0}, respectively. Moreover, everyquasiconformal self-mapping of C is homotopic to id. Hence ?(C) consists of asingle point. Next, every quasiconforma.l self-mapping of C - {0} is homotopicto id or to the conformal mapping z e lf z. Hence, fG - {0}) also consists of

a single point.

Finally, in the case of a torus .l?, we can show by the same argument as in $2of Chapter 1 that ?(R) can be identified with the upper half-plane .EI.

'7'I ureroeqJ o1 spuodsarroc euruel 3urmo11o; aq;' t-! .t ! .n

r.7 sserdxe osp e1y rJ / H = Spue 'rJ dnor3 uersqrnJ reqloue oluo J;o Jg ursrqdroruosr ue elsq e&r uaql

'J)L 'r-!oLo!=G)!g

^q PeuseP $ qcrq^r

'(a,'z)rsa * 1:!6ursrqdrourouroq airrlcalur ue aA€q a r '/

UI Ie?ruoue, aq1 Sutsg'tr o7 Tcadsat

qryn I to l,ttlu lonuouw eql / qqt II€c a1l ('g ot g;o Surddeu leruroJuoersenbB Jo uorl?rJlser aql s.r prr€ 'flanbrun pau[uralap sr 3f ue qcns 'gg'7 uorlrsodor4fS) '- pue 'I 'g;o qcea saxu qcrq^{ ,g *- U : ,f Surdderu l€ruJoJuocrs€nb e;o

H +- II : / lJll eql reprsuoc s,{e,rnp am 'asrmreqlo al"}s lou op e,ra JI '{p!} - J Jo

?uaruala auiosSo lurod paxgs sr oo pu"'I '0Jo rpso l"ql aurnsse feur ea,r'mo11'02'Z eululerl ,tq tf Jo slurod

paxg eqlJo rlc€e tuo.r; lcurl$p sl zlJo lurod paxg /tue uaq?'eqoq.radfq s-t 'I,L fes'1uaua1a auosJI 't={{!Llyo flurtlelnuruo?-uou ,tq lcurlsrp fgenlnur are t=j{ fa}

?eql eas uec aal'f,LJo turod paxy enbrun aql eq fd Surllel ueql 'cqoqered ere f,L

IIe JI '(0I'A pue g'U ssrutueT '3c) aqoqered .ro cqoq.radfq raqlra sr f,L frerra letll

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'slurod ee.rq1

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dno.rg uursqrr\{ u 5o acudg roilnurqclal 'Z'1-'g

'(*+'t) Ie^ralur

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'1urod alEurs " Jo slsrsuoc ({O} - V)1, p* (V),1

Jo qlea eruag 'pr o1 crdolouroq q U;o Surddeur-;1es Fr.uJoJuoctsenb frarra 1eq1pu€ 'U o1 lualerrrnba ,tlpurro;uoc sr Eurddeur leuroJuo?rs€nb e itq g;o a3eurl aql

leq? eas ot llneurP 1ou s! 1I'to) - v =a tol = u ?"ql asoddnS '6 a\du'oag

'{(* >), > lrl> I I C ) z} surcurop Sutr.ro'{O} - y'(qsrp lrunaql) y Jo auo o1 luele,rmba flpur.ro;uoe sI U JI ,t1uo pue Jr u€Ileq" sr g ;o dnor3

IelueruepunJ eIIl ?sqt 6 laldeqg Jo U''$ m I {retudg uI Palou aA"rI a^r 'es?c slrllu1 'aue1d-;1eq .reddn eql 'Il = U. ler{t arunsse sfea,lle e^a 'uo araq uro.r;'snq;

tzrsaezdg re[lurqf,ral Jo uorltrnrlsuog rr1,{puy '1'9

I22 5. Teichmiiller Spaces

Lemma 5.L. Two points [Sr,.fr], lSz,fz) € "(n) satisfy [Sr,.f i] = [S2,f2l inf@) ,f and only if 0i, = 0i", where fi is lhe canonical lift of fi for eachj ( = I , 2 ) .

Proof. Finst, suppose that [S1, ft] = lSz,/z]. By composing a suitable conformalmapping of ,S1 onto 52, we may assume that .S1 = Sz, and fi is homotopicto f2.A homotopy between .fr "nd /2 is written as a l-parameter family {.fr}r5r5, ofmappings of l? to 51 . Let /, U" th" canonical lift of fi with respect to f . Thenthe homotopy {ft} bas a unique continuous lift, say {F1}, under the conditionthat F1 - fi, and {fl1} gives a homotopy between fi and alift F2 of f2.

Fix an element I e f and z € Il arbitrarily. Then both of the paths {f'1 o

{z) | ! < t < 2} and {it " t " i;r1rr1z71 | 1 < t ( 2} have the same initialpoint f i o7(z), and have the same projection {f, I L < t < 2} on 51. Hence,both paths actually coincide with each other. In p-arlicular, the terminal pointF2"7Q) of the former is coincident with hoto ir '6'rQD. Since z is arbitrary,we conclude that

F 2 o 7 o F ; t = 0 i , 0 ) .

Since 7 is also arbitrary, and since each of 0, 1, and oo is fixed by some elementof f -{id}, we see that F2 fixes 0, 1, and oo. In fact, assume, for instance, that 0 isthe attractive fixed point of a hyperbolic element 7s. Then F2"1so Flt = 0 i,(to)is also hyperbolic, and has 0 as the attractive fixed point. Hence, we see thatr2(0) = 0.

Thus we have shown that F2 is coincident with the canonical lift f2 of f2with respect to f , and hence 0 i,

= 0 i,.Conversely, assume that 0ir- 0ir= d. Then, for every 7 € f we obtain

i i o l = 0 ( t ) " i i , j = 1 , 2 .

For every t in the interval [0, 1] and every z € fI, letting !" be the geodesic

frith respect to the Poincard metric) connecting fi(z) and fz(z),we_denote by

f ( r , t ) the point which d iv ides g, in the rat io t : (1- l ) . Then { f i = f (z , t - l ) |1 < t S 2] is a homotopy between fi and /2. From the above, we have

f t o t = 0 ( t ) " f r , I e l , t € [ 1 , 2 ] .

Hence, "n".y /, is projected to a continuous mapping fi of Rinto 51 - ^92, andwe have a homotopy between f1 and f2. E

Noting this lemma, we set

f#Q) = {ei l/ ir u .utronical quasiconformal mapping of C

such that eiQ) = if i-t is a Fuchsian group).

We call this "#(l-) the reduced Teichmtiller space of l- . It can be also re-garded as the set {d;(f) l0; e T*(f)} of Fuchsian groups equipped with iso-morphisms to l-, or lquivaleirtly, the set of "marked" Fuchsian groups obtainedas deformations of f by canonical quasiconformal mappings of C.

eql ul se / qcns fraaa ro;'lxaN'I'9 €uruerl fq ar'r1cafut pue'paugap{le^r sl

a^oqe se G)+,t> !0 ol (2I)J I U',g] tutoO e spues qerqn Surddeut aqa'!oo.t4

'(.7)t rttp^ pa{t'7uapt oslo sN (a)l,urrlt'Tcodutoc s? A fi'.t'aqTtng'(sles so) (,t)*Jyr?n pe{lwew s? a Io (d1 acods.ta11nu'w?eJ eql

ueqJ 'A aao!.tns uuDutaty o to Tapotu uocsq)nl D eq J pI '8'g uollrsodor4

:3ur,no11oy aql apnl)uot e^\ snql

tr'Uuo c! -

V r"qr aos e,rir'frerlrqre u ) acutg 'Q)"1 = ())! wUt apnlcuoc utsc a^r

'(oz)!! o ("L)e - (oz)uLo ll

acuIS') uorlrun; llr€lsuoc e ol I/ uo r(prro;run {1eco1 se3la,ruoc t?{",t} ecuanbas

e qcns leql saoqs 8I'U eurtuarl;o ;oo.rd eq1 'rerroa.rot{ ') = (oz)"t -*'tu-Il

leql qcns J ul I=J{"f} ecuanbas s slslxe eleq} 'U I ) ttla.te .ro; '61'6 uotls

-odor4 r(g'g J 0z lurod e xld'd ='!9 = )!,e WW asoddns'f1asra,ruo3

''! g - '! g susatu y?lq,ll 'J 3,L freaa .rog

"!6 -'!g 'ecue11 'u uo reqlo qt€e qll^{ }ueplf,ulo, arc (L)'!g pue (r)Yg suol}eru-roJsu€rl snlqgl tr 'J ) L frarra ro; ueqtr 'U uo z! -

V teqt asoddns '1srtg /oo.l2'

'vuo z! =r! fi fi1uo puo !? "!e='!6 frlstlos (7,'l= f) lS *A

: I s|utddoul lotn.totuousonb om1 'Tcodtuoc s, A 7oq7 asoddng 'Z'9 BtuuraT

:3ura,l,o11og eql a^€q ear 'f1en1cy 'e?eJrns uu€uelg

pasol, e q J I H - Ar ueq!\ (t) +J qll^{ luaPlculot sI (J)J leql a}ou altl 'are11

'(,f)Z f" aceds pacnpar Jo PUI{ € sI

(D+J 1eq1 des uec e^a 'acua11 '(t)*J1o lurod aur€s eql osle a,rrE (.i')13o 1ulodetues eql Surururrelep s3urddeur o,lrl l"ql r€elf, sr 1t'4reureg e^oqe eql tuoq

('fem eures eql ul J dnor8 uersqcng ,t.re.r1rqrs lre roJ PaugeP eq uec (.7)7

aceds rallnurqclal eql 1eq1 elop) '1 1o acods relptaq?Nal eql (J)J II€c aA\

'{Q)cbe-l[rn]]=Q)t

les 'rn Joss€lc acuel€Arnba aq1 [nr.] fq elouaq 'lI uo eot = rlt JI reqlo qcee ol Tuapatnba ate

Q)Cb ) zm'rm sluetuale oA{1 }€tll ,tes a6 'sdno.r3 uslsqlnJ osle are r-rnJrn }eqlqcns C ;o ar s3urddsu IeurroJuoctsenb lecluou€? IIe Jo les aqt eq (l)CCI l.l

'sa\olloJ s€ J laPoluuersqf,ng aql Jo (.7)"1' aceds re[nuqcla; aql eugap e^{ rPu€rl raqlo aq} uO

'(6 ,ra1deq3 ul 8't$

Jo Z {rsrudg ur uartS s?^\ uolllugap esoq,r,r) J Jo (U :t) ("f)Z }as lIruIT aq} uo

z! = rlgr,{po pue y. "!6- V6 r(;sr1es (?,'l = D lS r-g : f s3urddeur leur.ro;

-irocrs6nb o Ll l"q? 'rnolaq

U'g eurual 3o;oo.rd eql q se 'noqs usc aM 'tlrDureg

tzrsaoedg rafilrurlf,ra.1 Jo uollf,u]suo3 ctlfpuy '1'9

124 5. Teichmiller Spaces

definition of T#(R), we set \ = gi(f). Then /is projected to a quasiconfor-mal mapping.f of R = H/l onto'^9 - H/\, and hence determines a point

[S, /] e ?(.R). Thus the original mapping is also surjective, and we have the firstassertion.

The second assertion follows by Lemma 5.2.

5.1.3. Teichmiiller Distnnce

We shall now introduce a topology on "(.R). For this purpose, we define a dis-tance on ?(,R).

lake a point [,S, fj e T(R). Let py be the complex dilatation of the canonica]lift / of / with respect to l-. Then we have

0 i Q ) o f = i o t , ' r e r .

Hence, for almost every z € Il, it follows that

10i0)' " f) . f" = (f, o t). t'

( o i 0 ) ' " h . i , = ( i z " i .V .

pi = jti " t)|/l a.e. on ff, t e f .

and

Thus we obtain

(5 . 1 )

Conversely, if (5.1) holds for every 7 € f, then we can see that 0i0) =

i"l"i-'is a holomorphic homeomorphism of 11, i.e., belongs to Aut(H). Hence,we conclude that d;(l-) is a Fuchsian group, which implies that / is projectedto a quasiconformal mapping of .R onto a"nother Riemann surface H/0iQ).

We call a bounded measurable function 1t on H satisfying (5.1) with p insteadof py a BeVrami differentialon I/ with respect to f. We denote by B(H,l-) theset of all Beltrami differentials on 11 with respect to l-. F\rther, we set

B(H,rh = {p e B(H, r) | l lp l l - < t } .

We call any element of B(H, f)r a Beltrami coefficienl on H with respect to l- .Simila"rly, we call a measurable (-1,l)-form p = p(z)d//dz on R such that

llpll- = ess.supz€Rh(r)l < - a Beltrarni differcntialon r?. Denote by B(R) theset of all Beltrami differentials on ,R. Further, we set

B(R) '= {p e B(R) ' I lp l l - < t1.

We call any element of .B(^R)1 a Beltromi coefficient on R.

Remark /. By the definition, B(.R) and B(H,f) are canonically identifiedtogather with norms.

Also, for every quasiconformal mapping f of Ronto another Riemann surface,the complex dilatation Fi e B(H,l-)1 of the canonical lift / of / determines

1eq1 sarldurr p Jo uorlgsep eql '0 < I f.ra,ra .rog 'P aruet$P Jellnurq?Ial eql

o1 lcadsa.r qty( (lI)J ul r1:{[V ''S] = ud] acuenbas fqcne3 krc a4ea 'too.r'4

'ecualsrp rallnuty?tal ay7 o7 Tcadsa.t ql.tm

aTayihuoc sy (os1o (,t) *J acuaq puo) (g)a acods reIInu'UcNU eVJ '?'9 tueroaql

'y {sp }fun aq1 uo ecu"?srp arccurod aq1 sr d araqar'(g')pO) 6't {rare ro;

(6d ' Iil)d df,' 't'"=

I Atilln-rl _ -\ / (lttA- rl . ,\ ) Ht \l u,r - {,t I t) I VAlt)}3o1dns'ssa=

(,-ro'r)v3o1

1eq1 saqdurr 91'7 uorlrsodord'6 ,lJDu?A

'n raldeq3 '[gg-V] olqel elus]sul roJ aas '.1 dnor3 uetsqcng preuaE e

Jo (J)J uo r(3o1odo1 e roJ 'uorlsrgrluepr qql repun (U),2 f" leql ,(q PeugeP sI

("f),2 oo f31odo1 e '(t'9 uorlrsodo.r4) (U),2 qt1,r.r PeUIluePI sl (J)J a?uIS 'acuel

-srp ra[ntuq]lal qqt ,tq (u,)-t uo f3o1odo1 e eu$aP errr 'lcedutoc sI Ur UaIIA\'zd = rd 1eq1 saqdut sq;

'J ) L 'r - (.t) '-!!"t e

'N I u'J ) L'(L)'!e=(L)'-!:'{oe?urs 'pu€q

raqlo eql uO '.& uo .,(lurro;run f1pco1 pr o1 saErarr,uot uQ lsql 98'7 uorlrsodor4

urorJ s1r^olloJ l!'oo 3 r, s" 0 - "0rl e?uIS'u,tra,ra ro; Q)'lg o1 lcedse.r qlrrlr"f

Jo IJII Iecruouec eql aq u0 1:r.-l 'oo - u se I - ("6))I leql qcns zt'r{1 u1

t;:{"f} ecuanbes € sr areql leql e}oN '0 = (zd'td)P tet{t asoddns',t11eutg'(26)>t

.(4X ) (0" t6')y teqr l)eJ eql ,tq sno11o; flqenbaut a13uetl1 aq1 '?xa1q

'crrlaruurfs sr p eruelsrp Jellnuqlletr aql terl? I'g sururarJ ;o;oord eqt ul

se luaurn3re aq1 Sutsn fq aas u?c a,ri\ '1e.raua8 ul (t-t))I = (6)X ecurs '1srrg

'ecuelsrpJo suorx" eql sausrl€s uorlcunJ sII{l }eq} {caqc o1 oaau aa.zd

pue IdJo se^rleluesa.rdar eqlJo etlol{f, al{lJo luapuadeput { (zd'Id)p pqt aas

o1 fsea s-r tI 'zd pu" Id uee!\laq (U),2 uo e?uDIsrP r?nntuq?try eq1 flrluenb stql

IIe? a11\'(f .lo tltt e Jo l€q1 ''a'l) dJo uolteteilP Isurxeru eql slr€atu (6)y 'arag

'(f)y so1 z1' tt r)6 - (d' ril)p

las eA{ ' FV o zI

o1 crdoloruoq ere qclqa zS' otuo rg ;o sEurdderu IsturoJuortsenb IIe Jo las eql

eqz['tI1 ]al'(U)J ) lzt'zg1 = zd'lrt 'IS] = Id slulod o,r,r1 fue ro3'lltop

't 1o Tuatc$eoc

9Zr

'{d fq lt e}ouaP PUB,tuo4pg aq1 r/ slt{t IFc e^t 'I(u)g 3 r/ lueurala ue f11e.rn1eu

13ql aPnlluoc el\{

sacedg raflnruqf,Ial Jo uollrnrlsuo3 ctp{puy '1'9

126 5. Teichmiller Spaces

we can find a sufficiently large N. such that, for every n, rn ) N., there is aquasiconformal mapping, say fn,^, homotopic to f^o fi-l and satisfying thatl lp",-l l- ( e, where pn,^ = pl..^.In particular, we can find a subsequence{l"r}pr and a sequence {fni,n;+, }p, of quasiconformal mappings such that

l?tnr 'n,*J l - < z- i , i = r ,2,3,""

Next, let ps be the base point of T(R). Since {d(ps,p")}Lr is a boundedsequence, we may assume that K(/r) ( 1( for every n with a sufficiently large1( (> 1). Since

1 L r - jK ( f n , , n , * ) s # < r + 4 . 2 - i

for every j, we see that

g i = f n i - r , n ; o f n i - z , n i - r o " ' o f n t , n , o f n ,

is a quasiconformal mapping of R onto .9,.r, homotopic to /r' and satisfies

i - rK ( s i ) 3 K . l l ( 1 + 4 ' 2 - i ) .

j = 1

Hence, {f(gi)}r_4, is a bounded sequence. We denote by Kr the supremum of{rcki)}.

Now, let ii be the canonical lift of gi with respectFj = li i; belongs to B(H,f)r, and llpill- ( &1 = (1Also, we have

1 " " " r r - l l P i * ' - P i l l, l lui- rr+rl l- sl l l :F;r,,.r l l_ = l lru,," i+,l l- (2-r

for every j. In particular, {pi}p, is a Cauchy sequence in B(H,l.). Hence,pr = l imj**ti exists in B(H,f), and satisfies l lpll- S et.

Let / be the canonical p-qc mapping of 11. Then we can show that / belongsto QCQ). Let p = [S, /] be the point in "(.R) determined by d1. Since

,^^ , (d(p" , ,p) \ - l l p- p i l l I , , , rtann I -\ ;:)sllT:wll- s Tl6yttt'i- r'tt*'

we see that pn, converges to p. Since the limit of a Cauchy sequence is unique,p, also converges to the same p. This implies the completeness. tr

Now, fix a point lPa-1_,ftl € T(R) arbitrarily. By setting

[.f,].([S, /]) = [S, f " f{r], [S,.f] e r(R),

we can define a mapping lf1l. : T(R) - "(Rr) of "(E) onto the Teichmiillerspace ?(.R1) with base point [R1,fd]. Moreover, we have the following proposi-tion.

to l- for every j. Then- K')10 a Kr) (< 1).

'lertuereglp alprpenb crqd.rouroloq pell€r-os € Jo e^Il€luesardat e se

pereprsuo? aq uec urroJ stql ul a(r?) ,t1r1uenb aq1 '.le1e1 ureldxa II€qs a^\ se 'e.la11

| . ., Iltutz\'qlt, + )pl "l,,tl lJr 20q)

' l''

zl

sauroceq zlzpq + zpl w1t ees a^r'()),t = z Eurddetu l€r.uroJuo? e fq z ralaure.red l€?ol eql 3u€ueqc uodl'.leP{+ zpl zklfi + )/)) uroJ eql seq / repun aueld-ar eql uo .lrnpl ctrlau ueapqcng

eqlJo {teq-1nd aq1 '(t + >f)/(t - N) = { les erlr uaqo,'acue11 'I< >I auos roJ

z z

a, J--frp+ex<---12 -r-)I' l+>I

uroJeqlul ?)t allr^t uec aarr. 'aue1d-rn eql o1 (Eutsse.rduroc .ro) Eurqclerls Pue uollelorelq€lrns pue'aue1d-z eqt of uoll"lor alq€tlns e 8ur,t1ddy'9 3o Sutdderu eugeSur,rrasa.rd-uort"lua-Iro ue aq (15/[ < lol'C > d'o) 4d * zn = (r)l - n Io"l

'716 snuaS;o as€c eql lo; s3urddeur ((lecluouse,, rBIIluIs ssn?$p fleqs eA\'3;o Surddeu aulgp Eut,rrasa.rd-uolleluelro u€ Jo uollf,elord aq1 rapun

I; ur lurod raqlo rtue ol lues sr'1 snueS;o aceds reilnruqcletr, aqt'U ur lurodfue lerll uees eleq s,lr 'auo snua3 go sa?€JJns uu"tuerg pesolc '-t.ro1

;o asec eql uI

slerluara.Dlq cllBrpen$ crqd.rouolog'I'Z'g

srrroroaql s6rallntuqclol pue sturddel4l rallnuqr.ral'Z'9

'1 raldeq3 ur paugep esoql qlla luePlrulocaJ€ uorlres sql ul PeuueP

6J Pun (g),, t*qt uollces lxeu aql uI ^^.oqs oA\

'k?) 0 snual lo acods;a11nuqz??J eq11-I II€c Pu€'t;.'(q aceds

e qcns alouap feru a,s. 'ecue11 '1urod aseq aql Jo luepuadapur $ Ur qcns ro3 aceds

rallnuqcrel aq1 '{laurep '(27) 0 snue3 atues eql Jo U sac€Jrns uu€tuaw pesolt

ge ro3 crqd.roruoeruoq fpenlnur are (g)g 1eq1 saldu1 g'g uorlrsodotd'6 qroueq

'(g); aceds rallntuqtlal er71 1o Tutod asoq

aqT to uorTolsuvrt e (rA)t * (U)Z : *[t/] srql se qcns Surddeur e ilec aiA

tr ..,l.r1aurosr ue sr _[r/] ]erll Jeal,

sr lr ocueH 'r-rto6'r-r{o/^/ qtl^ seplf,utoc e?uelslp rellntuqclel eI{} Jo uol}IugeP

e'41q 6'tl rtlT,o*J eql '(U)J I [f ',,S] =b'll 'S] = d slurod orrtl fue .ro;']xaN

'uorlcafrq € sl -[V] fpealc'-[V]Jo Surddeu esreAul eq1 sar-r3 (U),-f -- (,U)-f : -[r-VJ ecurs'1srrg 'too"t4

'(a),f, ot nr1ifu,ou,oatuott s? (rA)J'.topcr,7.tnd u1'se?ullsrp rellnruy?t4 aq7 o7 Tcadsa"t qTtm utstyd.t'ou.to

-euroq lvrNrleuros, uo n (rg)a - (A),t : *lrll |utddDur styJ '9'9 uorrysodo.r4

LZIsueroeqJ s.ralFurq)ral puc s8urddel4l rallnurq)ratr 'Z'9

I28 5. Teichmiiller Spaces

A family p = {pi} of holomorphic functions pj on zi(Ui) for all coordinateneighborhoods (t/i, zi) of a Riemann surface ,R is called a holomorphic quadraticdiffertntial on .R if it satisfies

p* ( rn ) = p j o z1 r ( zx ) . ( z1e ' (21 , ) )2 on U i f iU3 ,

where zi* = zi o zk-r.We express (5.2) simply as

(5.2)

e*Qp) = elQi)(lzi /dr*)'.

We also writeg = 9e)d,22.

Denote by A2(R) the complex vector space of all holomorphic quadraticdifferentials on r?. A holomorphic quadratic differential corresponds to a holo-morphic automorphic form of weight -4 with respect to a Fuchsian model l-of .R acting on the upper half-plane 1/. Here, a holomorphic automorphic formp(r) of weight -4 with respect to ,f is, by definition, a holomorphic function9Q) on Il such that

p\eDt,e), = e(r), z € H, 7 e r.

We denote by A2(H,l-) the complex vector space of all holomorphic automorphicfunctions of weight -4 with respect to f .

Remark. From these definitions, Az(R) is canonically identified with ,42(I1,f).In fact, any element of A2(H,,l-) clearly determines an element of ,42(r?). Con-versely, for every p = {pi} e A2(R), formula (5.2) implies that the family

{piQi o r)((21 o r)')'}, as a whole, determines exactly one single-valued holo-morphic function on I/, which belongsto A2(H,f). Here, r: H - R= H/fis the projection.

5.2.2. Teichmiiller Mappings

As a "locally affine" quasiconformal mapping of .R, we take a mapping / suchthat for some constant &(0 < & < 1), it satisfies

fz = kf"

for a suitable local coordinate z around almost every point of -R. More precisely,we discuss a quasiconformal mapping / whose Beltrami coefficient 11 satisfiesthat

w = k &le lwith a suitable p e Az(R).(See Proposition 5.19 below.)

Let a positive & (< 1) and I e Az(R) - {0} be given. Then we call aquasiconformal mapping f a formal Teichrnilller mapping of ,R for the pair (ft, p)

t"qt ilecer 'oqy 'arrlcefu1 pue peugePlla^{ q 34i leql f1dutl 96'6 tueroeqtr, Pue

96'6 uorlrsodord 'I'9 "ruuarl 1"qt eloN 't=[{(llgD.l '(tfy])Y} = (S).1eraq^r

'(a),t >ll'sl '[(r)Y's] = ([/'s])3o,(q ue,rr3

o,!J * (a)1, so

Surddeur e a^"q aa,r'g, uo t=[{[t7]'llV)] = 3' Surrgeur € Eqrxg (pueq .raqloeqt uO 'ursrqd.rouroatuoq a rlrefrns e sr oI ,-

orlJ, oJ ?eql os 6 raldeq3 go g$

ur. o,!J uo f3o1odo1 e ernporlul e { l€q} pus 't'I tueroer{I uI Palels n ,r"(A)JIIll^{ pegrtuapr.4 pl"J teql IIsctU

'g 3o sursrqdrouoaslP Sur,uase.rd-uolleluelro

Sursn ,(q 1 .ra1deq3 Jo g$ ul paugep ?sq1 q pp(A),t Pue '(e ?) f snuaE 1o sacsJrns

uusruarH pe{rcu pesolc lle Jo tes aW sl priJ 'fleurep 'ra}}sl eql roJ o,!J pu"

pp(a),t uollelou eql e6n ea,r 'uotlaas slllr uI '1 raldeqS ut PausaP esoql qll^{ PeUlit,r"pl are .raldeqc slt{l w pel?nrlsuoc 'J put (U)Z r"qr ^{oqs 11"qs em '1srrg

'6 raldeq3 Jo g$ ul peugap tg

aeeds a:1cr1E eql esn aal 'asodrnd $q? roJ 'ursrqdroruoauroq arrtlaalrns e u (gt)J+ t(g)zy : 7 Eurddetu srrll leql ^toqs ol $ uorlces slqlJo esodrnd uleru eq;

'0=6

to! p.r - / pue'0 I dt tolEutdderu re[nuqclal € sl (U)/ =,9 - g' : 3f araqm

tt(g)zy ) dt '$'Sl = Qt)-f

(U)-r *- r(g)cy :1

fq paugap

Sutddeur € eA€q am ',uo1q

('leuorsueunp auo sr (A)"V'g' snrol e Jo es€e eql w l"tll ilersg) 'aceds

rolf,el xalduroc leuorsueurp-(g-rg) e f11en1ce { (U)zy teql s[el rueroaq] s.qoog-uueruerg 'laloeto141 'tll . ll ulou slql qtl/'t aceds qeeueg xalduroc ts sE ParaPlsuocq (U)zy pue '(U)e!. ) dt f,ue roJ ellug q llldll 'esec stql uI '(Z

<) f snuaE;o sace;-rns uueruerH pasolc Jo es"c aql ,(po rePrsuoc eal 'uolssncsrp Eurmollo; aql uI

'dt ro1 |utrltlout rqlnutlcyaa e (dt '1) .rred qql roJ Eurddeur railntuq?Iatr,

I€nrroJ e IIec e^ytlldll = { leqt arunsse fetu am 'r(A)zV 9 d luaurala ue rod

W\uil se flduns uelllr^{ ualJo sI 1e.rta1ur srql)

'npxpl(z)d>l

1nd eru '"zp(z)dt = o5 Eurltraa 'uo araq uro.rg

.{r > Illall | @)"v ) 6} = r(a)zvtes

qlrm dc dq o5 aaeldar e,,n ueq,o paiueqcun q ld',lldttrto; lffolt:;"Jtif;:";asec aql o1 puodsarroc q?Iqa 'sEurddeur Jallnurqclel leuroJ se oqe s3utddeur

I€ruroJuos preSar an'ara11 'Vllaq o1lenba sI /Jo /r/ ?uapgaoo rurerlleg aqlJr

"[[z='llall

6Zrsuaroaql s.rallnruqtral put sturddel4l ra11ltnq]ral 'Z'9

130 5. Teichmiiller Spaces

the identification betweenT(R)'td andTftd was given by the same @s. Hence,@e is clearly surjective. Thus we have the following lemma.

Lernma 5.6. The mappingsiDy:T(R) - - - - - -T; 'o and FsoiDy:T(R) + Fg arebijectiue. In particular, Fo = Fo oAy(T(R)).

For the sake of simplicity, we set t = fs o(Dy o 7. Then we obtain thefollowing:

Lemrna 5.7. The rnapping t : Az(R)t - Fc is continuous.

Proof. Let {p"}f;=r be an a.rbitrary convergent sequence in A2(R)1, and rps beits limit. For every n,let fn be a Teichmiiller mapping of .E for gn, and fn be thecanonical lift of /. on fI with respect to f , where f is the normalized Fuchsianmodel for [,R,I]. Set

f ^ = i , f i ; r , n = 0 , I , 2 , . . .

Then t, =i(g)_is a point of 1, representing /l. by definition.

We set lrn = in o /;1 and Fn = pi1^ for every n. Then we obtain

, , = ( r ' i ^ - ' i , @ ) o i ; r .\ '- l 'r. ' Fi" (fo)" /

Since lim,,*- llp"llt = llpollr ( 1, we can find a positive I < 1 such that

l l p " l l - < r ( t = 0 , 1 , 2 , " ' ) .

When gs = 0, then lim'-- l lp"l lr = 0. Hence, by Proposition 4.36, f, ' .orru"rg".to id locally uniformly on I/. Even when gs f 0, we can show the same assertion.In fact, since lim'-- llp"-pollt = 0, Qn(z) converges fo tfisQ) locally uniformlyon f/, where Q^Q) is the element of A2(H ,l-) corresponding to rp,. Hence, lettingH' = {z e H I ti'sQ) I 0}, we can show that pr, converges to 0 locally uniformlyon Ht, which is enough to show the locally uniform convergence of {h"}[1 on.I1. However, since it needs a fairly long argument, we first finish the proof ofLemma 5.7.

Since f,, converge-s to id locally uniformly on I/ in any case, i- " t " f;tconverges to iso"l" i;L for every 'l e l, which implies that t, converges to ts.Thus we have proved the assertion.

Now, we return to the proof of the locally uniform convergence of {i"}p,to id on.F/ even when gs f 0.

For every n, we set

I u " Q ) , z € Hv " ( z ) = \ 0 , z € R

I t l a , z e H *

tr '(t)".t l?)"t = (r).ulsaop os a)uar{ pue'g uo r(pr.rogrun f11eco1 pe o1 saS.ra,ruoc uy roJ ",_{ uor}nlos

Ierurou eql 'y .reldeqC Jo g'U $ ur 6 f.re11oro3 fq ure3e 'acueg 'V - C uo 'e'e 0 ol

sa8rerruoc osle uy ?eql ees o1 fsea q ll 'C uo flur.rogrun ,(11eco1 p.r o1 saS.re,ruoc"r/' e?uls 'oo 1- u s€ y uo 'e'e

0 ol saEra,ruoc "y l€ql ux\oqs a^€q e.f$, snr1J'y uo 'e'e g o1 seS.reauo" ,_(Irl) o un 1eq+ epnlcuo? e/d

'{,H ) Z to rH ) z lV 3 z} uo fpuroyrun f11eco1 g o1 seS.reruoc un acurs'os1y'y uo sorez ou ser{ r(:"/) r(t"nn ecurs'y uo flurro;run,(1eco1 1o1 sa3raluoc

/ ,(t^l) \vQ'l)"{-r ,

\ ,\i,I) ) ,acuag .3 uo

flurro;run f11eco1 pr o1 saS.raluo, ,_(ynl) 1eq1 luaurn3re prspu€ls e fq rrroqs uecarrr 'os1y 'g uo fpure; I"ruJou e sr acuiq pue 'snonurluocrnba fgeool pue papunoq.{1uro;run ,{11eco1 { I?{r-(1"/)} ,(tg"l eql leq} eas uer eM'gi.,'V tuaroaqJ, Jo;oord eq1 uI pel€ls se,lr s€ '7 .ra1deq3 Jo (VZ'V) segsr?es r.,!' drerra aours 'alog

'3 uo {puroyun f1eco1 pr o1 sa3rearioc osp "r,f }eq} ,raoqs

Ileqs a A 'popunoq fpuroyun a.re uy

;o sl.roddns aq1 're1ncr1.red u1 'u d.ra,re ro;

(v)i^l -c)' 'o )

/ty"D \ )=(r)"Y (v)i^t >, ,r_(i^l)"\ffi-) )

ueqJ 'irl o "r/ se "r3f esodtuocap 'u fraaa rog 'lxaN

'p uo {pr.ro;run d11eco1 pr o1 seS.re,ruoc (z l1)ug lG)g = @)l^l1"q1 aes o1 fsea $ lr ueql 'oo ol spuel u se C uo fpr.royui pl "i se3rer'uocu4 tol 11, uotlnlos leturou aql '7 ratd€r{C Jo g'6$ ur 6 ,(re1oro3 fq 'ecueg '3

uo'e'e oo - u s€ 0 +- u/t pue u frara roJ { > -ll"4ll 'f1.rea13 .@/1)"a{lI

= (z):^t leql ^rou{ a/rr pue'y ur paureluo) sr urt,f.tare;o lroddns aq} uaqf

'v-J),

v)zlAS A^r'U Are^e rOJ

'1s.rl,{ '09't uraroeqtr yo yoo.rd eql ur s? .rjf esoduroeap arrr 'esodrnd srql .rog'p raldeq3;o

6$ ut pa1e1s suorlnlos l€urrougo serlredo.rd esn dluo aM roJ'qceo.rdde ,(reluauraleraql"r 1nq 3uo1 fpre; .raqloue e{el aal '1ce3 slrl} Jo yoo.rd e ua,rr3 1ou eler{ e,!recurs 'rela,!ro11 'flalerpeurrur uorlrasse eql urclqo eilr '7 .raldeqC Jo g$ Jo pue aqlle {I€Iuau aq} ul 1"3J eql esn eai!,Jr'aeua11 'c uo'e'e oo <- u sp 0 + u.l leqlpue u drarra roJ { > -ll"rll leq} aou{ a.tr 'os1y '(gg'7 uorlrsodor4 ;o;oo.rd aq}'Jc)

I/ o1 ? Jo ,r/ Eurddeur cb-un lecruouef, eql Jo uorlcrrlser aql $ "? uaqJ

e Qr=G)'l'u*;j = Q)i^

IIIsrueroeqJ s.rallnruq)reJ pue s8urddeyq rallBurqf,ral'Z' g

132 5. Teichmriller Spaces

Simila,rly, (but more easily by applying Proposition 4.36) we can show thefollowing lemma.

Lemma 5.8. The mapping f s o@2 :f@) -- Fe is continuous.

5.2.3. Teich-iiller's Theorems

The injectivity of t foilows from the following Teichmtller's uniqueness theorem.

Theorem 5.9. Let f he a Teichmiller mopping for an element 9 e A2(R)1,and letT(p) =fS,fl. Then eaery quasiconformal mapping h of R Io S whichis homolopic to f satisfes

l lpr, l l - > l lpr l l - .Moreoaer, the equalily holds if and only if h = f .

A proof of this theorem shall be given in $3, for it needs some preliminarydiscussions. Returning to the proof of the fact that T is a surjective homeomor-phism, we note the following corollary to Theorem 5.9.

Corollary. The rnappings T and t are injectiue.

Proof. By Lemma 5.6, it suffices to show the injectivity of 7. Assume thatf (pt) = T(pz) for some pr,pz e Az(R)r. Let /i be a Teichmiiller mapping forgi and.T(pi) = [Si,/i] for each j. Then the assumption implies that there is aconformal mapping h of Sr onto ,92 such that h o /1 is homotopic to /2. ThusTheorem 5.9 gives

llp,r, ll- = llttnoy,ll- > llp,t, ll-.Similarly, since h-l o.fz is homotopic to fi, we have

l lp r " l l - 2 l lpr , l l - .Hence we conclude that llp;,oy,ll- = llprrll-, which implies that h o h = fzagain by Theorem 5.9. In particular, F!, = Fiz.

Thus if pt = 0, then 92 = 0. If 9r * 0, then ll91ll1 - llrpzllr, and

n/lprl = pzllprl a'e' on R. Hence we conclude that 92/91 is positive a'e.on.R. Since pz/pt is meromorphic, it should be a constant. Namely, there is apositive constant c with gr = cgz. Since llrpllll = llpzli, we conclude that c = 1,i."., pt = 92, which shows the injectivity of 7. tr

Lerrrma 5.to. The image t(A2(R)) of A2(R)r under t : Az(R)r -------+ Fo is anopen set, and t is a homeomorphism onto its irnage.

Proof. By Lemma 5.7 and the Corollary to Theorem 5.9, we see that t is a con-tinuous injection. Since ,,{2(,R)r is homeomorphic to R6c-6, Brouwer's theoremon invariance of domains (Theorem 3.11) gives the a.ssertion. D

'(t@)"v)t)r-(soo o.d) = ((a)zv)-t = a

1aB aru '0I'g pue 8'g seurueT urord'-L to1 uorlress? eql ^loqs ol seclgns 1r 'g'g eurural 8ur1ou [.9'too.t4

.69 = (r(g)zV) [. ruo ,(A),r

= (I(U)zy) 1, 'Qauto71 'aatTcaftns ?ro -L puo -L sfutddpru ?ttJ .gT'g BrrnuaT

tr 'pe1)euuoc osle $ '.{ t*.{l ,tldtur g'g ptr€ g'g seurural '1xag'p91?euuoc asl/rrJI3

sl (U)Z snr{I'[.f ',S] prr [p!'lf] fuloa es€q aql uea^r?eq stceuuor qctq,rl (g)g ut

{t I l;0l[t/'rg]] a,r.rnc snonurtuoce ur"]qosaa ueqtr'@)rl = rg 1as a/vyr/lq ?uel?Ueoc rur€Jllag esoq!\ g;o Eurddeu l"uroJuorrsenb e aq T lel'I > I t 0qll,r{ 3 fra,ra rog 'd - Iil tas pus'(A),f,>'[/',S] futoa f.rergqre u" xld 'too.r4

'p?l??uuo? ero 6l puo (Ah secoils eUJ 'ZT'g BtutuaT

:3urao,o11o; eq? ilecal am '1srrg '7;o ,(1v'tlcekns eql r'roqs ileqs aal 'fgeutg

'snonurluoc q -t ?3r{} aPnpuos ein'{rerlrqre s o1 acurS'urErro eqt ts snonurluo? st t; snq;

'(oo - u) o -- +l4ll+ ?o1' ((4)ra'6)'t')pelsrl a/$ 'u fra,ra roJ (Illuf ll - I)/(Illutlll + I)

o1 pnba s1 u4l ro; Surddeur rallnuqrral e Jo uorlelepp Ieturxeur aql aculs 'oo

- ttr s€ 0 <- rll"/1ll t"ql q?ns r(ry)zv ul r?{ ",/l} eauenbas fue ar1e1 'nog'ur8rro eq? ts (sEurddeur rel1ntuqcrel Sursn ,(q 7 o1 ,{pelrurrs pauyap)

(-U)-f + t(rg)zy : 11 to1 enrl osle sl slql yr dluo pue JI a5 1e snonurluoc sr

1, 'acua11 'aEeurr slr oluo ursrqdrouroatuorl € s-r pue 'o1

Jo pooqJoqqEtau atuosur peugap{lam sr t(tgr)av + r(U)zV I Lo*lfl"r-(lr) = Io,-(!) t*Utsaqdur 0I'g surureT'(O)? = (a1)/. acurg 'trJo esec aql ur se fe,n etrrcs eql uI

t,tr - r(rg)zy :r1o ,!lU)o Keo ul = V

Surddeur " eugap eA\'f.rlaurosr arrlcelrns e q '[I/] f"ql g'g uorlrsodor4 fq ,raouq e^{ ueqJ '(U),1

Io Wg' tgr] lurod es€q eqt o1 d spues qerq,r '1urod es?q eql Jo (IU)J .- (g)Z: *[t/] uorlelsrrerl e reprsuoC'[V'IU] = (6)L - d 1as pue'fprerlrqre r(U)zy

3 d lurod e xg 'asodrnd stql rog 'snonulluo? q 7, lsql ^roqs ol sulsrueJ ?['3r uo snonurluoc sl r--r. ryqI ,(1dun g1'g pue

g'g spurrrurarl '(Keo ul) o ,_L = r-L a?urs '(r(U)zV)I = A uo paugapjla^r sr

r--L leq} s^{olloJ 1-t'6'9 ureroaq; o1 drelloroC aq} {q e,rtlcefut sry a?uIS 'loo.t4

'afputt,

s?, oluo rustrldlnutoauoq o q (U)J 1- r(A)zV : l,0utililout eVJ '1.1-'g BurrrraT

ttIsuaroeqJ s.ranlruqrlal pue sturdduq ralllurqf,reJ 'Z'g

134 5. Teichmiiller Spaces

which is an open set in ?(,R). F\uther, Lemma 5.12 implies that 7(r?) is con-nected. Hence the assertion follows if we show that the relative boundary 0E ofE in T(R) is empty.

Now, suppose that AE * 6. Take any [S,/] e d,E. Then there is a sequence

{p"}L[ r in ,42(R)s such that T(p") . * [ ,S, / ] , and l lp" l l r - - - - 1as n . - - -oo. Let /, be a Teichmiil ler mapping for gn. We set T(p") = [S,,f,].Bythe assumption, there is a quasiconformal mapping h, of S, onto ,S which ishomotopic to / o fil for every n such that llpl"ll- .-* 0 as n I oo. Inparticular, for a suitable & < 1, we have

l l P r " l l * < i , n = I , 2 , " ' ,

where g, = h;L o f.On the other hand, since g,, is homotopicto fn, Theorem 5.9 implies that

l lpr" l l * 2 l l t r t " l l - = l lp l l - - 1 (n - oo).

This is a contradiction. Thus we conclude that 0E is empty. tr

As a corollary to Lemma 5.13, we obtain the following Teichmiiller's eristencelheorem.

Theorem 5.14. For euerg quasiconformal mapping f : R - S, there edsts aTeichmiiller mapping homotopic to f

Lemma 5.13 finishes a proof ofproved the following theorem.

Teichmiiller's lheorem. Namelv. we have

Theorern 5.15. The mapping T: A2(R)1* "(Ii) is a surjectiue homeomor-phism.

In parl. icular, T(R) is homeomorphic to Az(R)r, and, hence lo Roc-o

In the course of this proof, we have also shown that all representations wehave considered as the Teichmiiller space of a closed Riemann surface of genus

S P_2) are mutually homeomorphic.

Corollary. The spaces f@), T(R)otd, Ts,T;td, Fs, and R6c-a are mutuallyhomeomorphic to each olher.

,tlluarcgns " qly{'{t > lrl > 0'!?, > z?rc > O I C a z} ureurop e sdeur ) srqlleql Jeplsuoc {eru ear'{errrfuy'0dJo poor{roqqftau fue ur panlerr-a13urs eq }ouue?

er(z+*)z ry = zlrd"onf = rD

?"q1ees aA\ 'acua11 'od

lo l) pooqroqq3rau etuos uo z eleutproot lecol elq€llns € roJ

'zP*z - 6

ruJoJeql ur uellrru. sr d 1eq1 pa,ro.rd q 1l'(I {) u^l lepro p 61o '0d fes'oraz e 1y

('61'g uorlrsodo.r4 oqe aag)

1eq1 uorldurnsse aql urorJ sl',rolloJ snll

.mllrr/ll =1,)tt +)= O)s,{q uaar3 g Eurddeur euSP u€ se

,{11eco1 paluese.rda.r sr a1 .ro; / Surddew railntuqcretr e 'seleurprooc-ol Sursn fg('sapurprooc IerolJo luapuedapur ele a1;o soJez leql lecag)

'(4 ur 'ro) 0d punoreaTourp.tooc-6 e Surddetu qqt IF? a7yyC o?q 72 ;o Surdderu leruroJuoc e sarrr8

2 > d 'zlt6

fq peugep uorlcunJ eq1 pue(od

, pooqroqq3tau auros ur qcuerq crqdrouroloq panl€^ e13urs e seq zp"1rQ)dt

"1rdt ueql'dt 1o oraz € lou sr A ) 0d JI'{0} - t(A)"V 3 d luaurele ue xlJ

IBI+uara.SrC crlerpun$ crqd.rouroloH B ,(q pacnpul r(rlouroag 'I'g'g

'fgatrq ((crtrleru, e qf,ns ureldxa aru. 'uorlcesqns lxeu

eql uI 'rtllncgrp ou qlr^r pernporlur aq u€c tl13ua1 pu€ €are s" q?ns suorlou eql'reae,no11 'dt go orcz.r(.ra,re 1e sale.reue3ap (f,rtr?eur,, srqS 'Surddetu Jallntuqcreluarrt3 aq1 o1 Surpuodsarroc I(U)uy Jo luauala eq1 x "zp(z)d

- o5 a.raqr* '9.

uo "lzpll(z)61 = "sp ((crrleur,, eql raprsuor ain 'crrleur e qrns sy 'Surddeur

Iellnluqcletr uartrS aql qll^{ pel?I?osse cularu eruos 01 lcedser q1r,n scrsepoa3raprsuoc ol lernleu aq feur l.r

's3urddeur rellnuqcrel Jo eseo eql ul uelg 'rrJ

-1eu ueeprl)ng aql o1 lcadser qll/'a scrsepoa3 e.re C uo seuq teql ileceg'Eurdderu

rellnurqclel e o1 lcadsar qll^{ (seull, ;o turueaur eql ssnrsrp aill '1srrg 'seur1

ol saurl puas ,{aq1 leq} sl s3urddeu euge Jo sellradord elrsrcep eq} Jo auo's3urddetu re[nuqoretr pa11ec sSurdderu,,aug:e

d1pco1,, 3ur,t1dde fq uear3 are ,,fcuercge,, lsaq aql q?!la seJnlcnrls xalduroc aq1

Jo suorl"ruroJep 1eq1 slrasse (6'9 ue.roeql) uraroaql ssauenbrun s.rellnuqcrel

uraroaql ssauanbrun s6rallntuqrlal Jo Joord 'g'g

#'=$'= "

"oof = "'

Jo=

98IuaroaqJ ssauanbrull s.rall$uqrral Jo Joord 'g'g

f36 5. Teichmiller SPaces

small r, conformally onto a "domain" {( e C | 0 < ""g( < (-* 2)t, 0 <

lClcZrt^+')lz/(rn*2)) spread over the (-plane. Hence, we may also call ( ag- coordinale around p6.

Now, we consider the "metric" ds2 = lgQ)lldzl2, which is nothing but thepull-back of the Euclidean metric on the (-plane by an arbitrary grcoordinate

c.To make discussions clearer, we consider the lift Q € Az(H,f) of I on the

upper half-plane ff with respect to a Fuchsian model I of R.For every piecewise smooth curve C on I/, we put

l,vQ)lu2ldzl.

We call this lCl,p the Q-length of C. For any two points 21,22 € I/, denote byL"r,"" the set of all piecewise smooth curves connecting 21 and z2in H. We set

do(rt, rr) = cr9!,,,"1c1a.

We call itlhe Q-distcnce between 21 and 22. An element Cs of L2r,7, is called at/-geodesic between 21 a\d z2 if lt satisfies

lColo = d,v(21, z2)'

Now, we describe how a r/-geodesic looks. Assume that there exists a t/-

geodesic C6 between 21 and z2 in H. For every P € Co which is not a zero of',i, tn" lenglh-mlnimaliiy implies that G should be a segment nea,r ((p) on the

(-pla,ne, where ( is a rlcoordinate, i.e., the composed mapping of a g-coordinate

and the projection of I/ onto.R. At a zerop € Co of rf of order n)0, C6 may

be broken. However, the angle at p should not be less than 2tl(m* 2). (See Fig.

5 . 1 . )

9 b = [J C

-y i

(lctlo < lCo l,p for 0 < 2r /(rn + 2))

Fig.5.1.

We call a closed arc -t on H a tf-segmenl if, for every interior point p of. L, L is

mapped by a r/'coordinate at p to a segment. By the definition of a tf-segment, it

'syutoil pua sp 6u4cauuoc crcapoaf-dt anbtun ?Ul s? ,I Tuau0as-dt y 'z(.re11orog

D 'UOllf,IP€rlUOC e SarttS qCrqar (zz .rorz ol Eurpuodserroc 1ou f, fre,la roJ (Z + !u)/t7 { fB ,ra,ra,nog .s,f aerqt ls?el1e rog (6 + !u)/vZ ueql ra1ea.r3 lou aq pFoqs !0 wql s,rrolloJ q'0 < N eours

t=!'k+ til"Z= (e(Z+ !ut) - "dZ

r€rrt apnrcuo, "^1"r.t"" 'luaur3as-d e uo

0=Qp?w)p7+Q)QEwp

leql 1lrou)I ellr 'raqllnJ

I=f'tty = (t6- ")T + Qparqpoef

er\eq ern 'pueq

raqto eql ug '^r(1rcqdr11nur Surpnlcul O q dt 1o sorcz Jo reqr.unu aql $ l.r ereqru

t-f

.tt = oef't0[ut'3+ rLNT, - (z)/t?wp

I

1€IIt ̂{oqs us? e^\ 'aldrcutrd

luatun3re eql ,(g'f,.,tre,ra roJ O u-r t+!7 pue f7 uee,lrleq a13ue aq1 eq (0 <)fd tat'os1y 'oraz aqol !s, irro1€ a^{ pu" I? - t*-I araq,u'f, frarr.e ro; r+!7g ll +n 4Jo sorez Jo rapro aql eq lur p"l'Q 7 u") t--,j{|il fq uraql alouap pue 'O p Oef.repunoq eql Jo uo.rleluerro elrlrsod eq1 o1 lcadsar qlr^r Japro us ureql e,rt3 e7y1'sluaur3as-dJo raqunu elrug sJo ls.rsuo? zC prre rC(eroJeq uees ueeq seq sy

'H ul CI ureruop ueprof e spunoq e, n I, uaqJ'{zz'tz} - zCl) Ig teqt r(lqereua3Jo ssol tnoqlr^{ erunss€ {ew au'.,(ressaceu;rslutod3o red e1qe1-tns e qlytr zz pue Iz Sutcelder ,fg 'alduns arc zC pu€ rC ',t1.rea1c

ueql'cz pue rz Eurlceuuoc zC'rC scrsepoeS-al o,rr1 ere eraql tsql asoddng ;l'oo.l2'

'anbtun s! zz puo rz futlcau-uo? crsepoe6-0 "rtt '

H ) zr'rz slutod l?urlstp omy fiuo rof, .gl.g uorlrsodo.r4

'crsapoaS-d e;o ssauanbrun 3uuao11o; aq1aleq elr 'os1y 'uraq1 turlcauuoa crsapoaS-q! s slsrxa araql 'Il

;o slurod oa,r1 f.ra,rero;'relncrlred u1 'acuelsrp4 sgt o1 lcadse.r q1u,r elelduroc q I/ teql noqs r(lseau€f, e^\ uerlJ 'u eceJrns uutsuaru pasolc € Jo es?? aq1 ,(1uo Jeprsuor aiu 'are11

'0 < ur repro Jo otez e 1e (Z + *) l"Zueq? ssel 1ou a13ue ue e{€ru slueurEas-d qens o^rl l"ql uaes elsq elr.r'ra,roaro141'61o sorc2 to'zz lo (Iz raqlla are slurod pue esoq$ sluauEas-ol Jo Jeqr.unu alrugs Jo slsrsuoc .g ;o slurod olnl Surleeuuoc crsapoaS-d fra,ra 'Suro8aro; eql uroJ.{'(t

> I 5 O) (l)z - z i I luaurEas-ol,tue Suop 116 olnpour luslsuoc e sr (uoeraq uorJ zzp@)QEre su uatlrr^r fldurrs q qclq,lr) "(l),r((l)r)fEre 1eq1 r"ep sr

1,8 IuaroaqJ ssauanbrull s.rallgruqf,ral Jo Joord 't'g

138 5. Teichmiller Spaces

Now, to prove Theorem 5.9, the following lemma due to Teichmiiller playsa crucial role. To state it, we prefer to returning to ft and rp. In particular, theg-Iength lll, of a curve L on R is defined by

l L lv= [v f , ' .J L

The projection of a rf-segment to -R is called a g-segment.

Lemma 5.17. (Teichmiiller) Lel h : R + R be a quasiconformal self-mapping of R homotopic to id. Then lhere is a positiue constant M dependingonly on R, h, and g such that

lh(L) l ,2 lL l* - M

for euery g-segment L. (Here, h(L) may not be reclif,able, i.e., it may happenthat lh(L)1, - x.)

Proof. Let h b" th" canonical lift of h with respect to the canonical Fuchsianmodel f of .R. Then it suf;Hces to find a constant M such that

li'G)lq >lLl,e - u

for every rlsegment L r" n.First , by Lemma 5.1, i t fo l lows that 7 'rol :1oh fo, every 7 € f . Hence,

letting C, be the rp-geodesic connecting z and h(z) for every z € H , we have

lC"lo -_ lCrpl lo

for every 1 e f . Since -R is compact, we see that

M = 2 s u p { l C " l , i l z e H }

is finite. (Note that lC"lq _is continuous with respect to z.)Now, let a rf-segment i be given. Let z1 and z2be the end points of i. Then

the curve C"r.ir1L7. Cr"-'aiso connects 21 and 22. Since i i. . rf-geodesicconnecting 21 and. 22, we obtain

lLlo s lc ",lq * li,G)lo * lc""lq s li6)la + M.

Thus M is a desired constant. D

5.3.2. Preliminary Considerations

A prototype of Teichmiiller's uniqueness theorem is the following Griitzsch'slheorem,which treats the case where .R is a rectangle {z = x*iy eC | 0 < c <r , 0 ( y ( 1 ) .

D 'l = rl 'sl leqJ 'p! - 6 WLllsaoqs l'? uotlrsodor6 ;o ;oo.ld ar{} Jo JIeq puoces eql ul se luaurn3re arues eql'g

;o secrlre^ [e saxg d ecurg 'S' Jo rorrelur eql uo l€r.uroJuo? ecueq pue 'cb-1 sr

,-t o rt - d lsql saqdurr qqtr 'U uo 'e'€ I = 'trt l€r{l aes e^\ snrll 'U uo 'e'€

l"(V)lt = l"(V)lPue

'l'(V)l+ l"(V)l = l'(V) + "(V)l

ul€tqo e^1, ueql 'sp1oq ,{1r1enba aq1 JI '}xaN

('g'7 eunual;o;oord eql qll,r.r a.reduro3) 'tl < rtl lerll ',(lluap^Fba

to'y = .rls < r)1 1eql epnpuor arlr arueH '(tq - t)/$t + I) = rX areq,$

,s. (.rrrr)l

np,pel,(!)t -,y$0"[ [

. uo,offi [ I t( ar f \|np,p(l'(t)l+ l'('/)l) I I lt "' z( JJ )

sa,rr3 dlrpnbeur (zre^rqcs 'z(tl) *'(!) ='(t/) acurg

'nprylasy11"U t, J J 1aB a'r'r

'[1 'g] .re.,ro f o1 lcadsal q]l^\ seprs qloq Surler3e]ul '[I '0] 3 n fra,ra 1sou1e ro;

opl@t+ ')"(V)l of

t Koul''t - @t +r)r/l = sil

leqt a?ou '1sug'too.t4

'l = V Il fi1uo puo lp snoq fiTtpnba ayy'taaoato1tg

'q<rq

7oq7 snopol l! '! puD 'p + s's'g o7 'fr1aatycedsa.t 't puo '? + r'r'0sdotu puo 'g

to .touaTu! ?q? ao cb-(tq - t)/Fq + I) sl q?lyn S <- g : ttutstyd.toutoau,oy fi.teaa .tot :fi1.tado.td purerp? 6utmo11ot ay1 sa{st1os t urrm

'I > (I + X)/ft- X)={'I <'t/s - X

pql peutnsso st 7t'a.tep 'g lo.touaTu, eql uo cb-y st, ycryn

'tu+sx=#=?)!fiq

pau{ap [t 'O] x ["'0] = g a16uo7ca.r pesop re1?ouo o7 f1'0] x [r'0] = U a16uo74N, pesop o {o 0utddotu Totu.toluoctsonb au$n uo aq t pI 'g1.'g uorlrsodo.r4

6tIueroaqJ ssauanbrull s.ralpruqrral Jo Joord 't'9

140 5. Teichmiller Spaces

Thus an affine mapping as in Proposition 5.18 is extremal. Returning to aclosed Riemann surface R, we find that any Teichmiiller mapping looks very

simila,r to an affine mapping. Actually, we have the following proposition.

Proposi t ion5.19. F i rg€Az(R\ arb i t rvr i ly . Let f : R+ S beaTeichmi l lermapping for g, and set k = l ltpllt (< l). Then therc etists a unique holomorphicquadmtic differenlial tlt on S satisfging the following conditions:

(i) tf p is a zero of p of order m, then f (p) is a zew of r! of lhe same order m.(ii) Let p be an arbitrary point of R which is not a zero of g, and ( be a 9-

coordinate around p. Then therc erists a $-coordinate u at f(p) such lhat

' C + n (u o f =L - n

(5.3)

We call cp and ry' in Proposition 5.19 the initial differential of / and the

Ieryninal differcntial of f, respectively.

Proof. For every p which is not a zero of g, we define a mapping u = up i\ a

neighborhood of /(p) by (5.3). Then in some neighborhood U of p, we have

Since p.o1 = kpllpl for every local coordinate ur on /(t/), we see that u ottt-r

is l.qc, and hence conformal on f(U). Thusar is also alocal coordinate around

f(p).Next, for a zero p of g of order rn, we have seen that g = z^dz2 with a

suitable local coordinate z. Define a, as a continuous branch determined by

, d c , pFro ! = tc a7

= E; i '@\ lY l

, " f = (z(m+2)12 I 1 ,2@+z) lz

)2t (n+2)1 - &

Then we can see similarly that c..r is a local coordinate in a neighborhood of /(p).Finally, consider (fu)' in a neighborhood of every point /(p) such that p

is not a zero of rp, where @p = u is as above. Then we can show that these(fu)'give a single holomorphic quadratic differential on ^S, which we denoteby /. From the construction, ry' clearly satisfies (i) a,nd (ii). The uniqueness of ry'

follows at once from (i) and (ii).

5.3.3. Proof of Theorem 5.9

Assume that the assumptions of Theorem 5.9 a,re satisfied. Let r! be the terminaldifferential of / obtained in Proposition 5.19. For every p e R which is not a

zero of rp, take a g-coordinate ( a,round p, and a ry'-coordinate r.r around q = f (p)

as in Proposition 5.19. Consider the "horizontal dilatation"

sff sff'"prp

lJ l4opr(b'6)y lJu-relqo a/',r '(g'g) o1 flrpnbaul (zreaqcs Eutfldde '1xep

(z.q) .a uo .e.e @)('!)r'x > ,(tolt:tv)l + (o)lr('/)l) ) ,@,r!)u

1aB a,n uaqa'r(6)lJ('/)l - r(o)l)('/)l = @)(riltpue'1-) 6rt o o= V'(I{ - i/$t+ I) = t)r''-ll'Irlll = r1 1as am'tsrl.{

aues aqr,(q o. s uaroaqr "^.,0 p ",8Id. ;iltff"Hi"i: i:lii ;iT'":.t:iT#'"r"0

fl t4op(b'qy [[o1 lualerrtnba q (t'9) flrlenbaur aq1 'aeua11

(g's)

sf f'tpop(b'6)y

I J

4op((b)r-!,'nutfl L= upy1a,'nu ll

aleq e^r 'bplpy - tpop acurs 'U J d ,{.rala lsourt€ roJ

(q.s) (d,tt)u = (@)1.6)yN

segsrlss pue'elqernseaur sf lI'S'uo'a'€ pausap u f;o

..1 oo Itollffil =(r'r)v

(uorlel"lrPleluozrroq,, eql ueql .!.t+o = r'l pu" ,_I ort = f 1a5 .frpxpl(z)dtl = bp?p areq/{

'tptpN"fl ,hp''p@'.ny ||

'trt * ? = )'tol 115f;;aal = (o'v),

:U Jo eJnlcnrls xalduroc eql Jo uorleurroJap ro; ,,fcuarrge,, lseq eqlseq / Eurdderu rellmuqcle;1 e l€ql lceJ eql luese.rdar o1 ,(e.n euo s? ,(lqenbaur3urmo11o3 aq1 preEar feur e,n '91'g uorlrsodor4 ;o ;oord aq1 Suqlecar 'no61'@ -

t)/Q + I) = l9r las air 'uorleas slq? Jo 1se.r atll uI 'U f d fre,ra lsourp roJ

ffi = (o) |1ojjy)sl = tor |#DI = (o'r).urclqo e,n ,o pue ) o1 lcedsa.r qll,rrr / lo (d,/)V .(uorlelelrp

Ieluozrrorl,, erlt roJ '1eq1 selldurr 6I'9 uorlrsodo.r4 leql II€ceU 'uorlcunJ alq"rns

-sarue s.r prrp'g uo'a'e peugap q (d'V)V slqJ'ra pue )ot leadsar qfl,r VJo

I?Iuraroarll ssauanbrull s.ralpruqf,ral Jo Joord 'g'g

t42 5. Teichmiller Spaces

Hence, (5.5) and (5.7) give

f f , - f f ( ^ $ r , d \ 'JJ,ooo' s JJ"(tf-, Kd(drt

t+ il "t (r,)(p) d(dr1 ='# | I,o,o,Thus, K1 2 K, and hence &1 ) /c.

Finally, if &r - t, then both equalities in (5.7) should hold. Namely,

l(/r)e + (n)el(o) = l(/r)e l(0) + l(/,).1(0)

and

l(/r)e l(o) = &l(/r).1(0)a.e. on -rR. This implies that py, = kQ/lpl.Hence, 9 is l-qc, i.e., it is conformalon R. Since g is homotopic to fd, the canonical lift of g on H is coincident with

fd by Lemma 5.2. Thus we have h = f .

Now, to prove Theorem 5.9, it remains to show the following lemma.

Lemma 5.2O. The inequality (5.6) holds.

Proof. lt is in this proof where we need Lemma 5.17.For the sake of simplicity, we assume that rltlz has a single-valued global

branch on S, which is a holomorphic Abelian diflerential on S. If not, take any

local branch of rrLl2, and continue it analytically as far^as possible. Then we

can construct a two-sheeted branched covering surface S of S, with a branch

point at every zero of t! of odd order,such that ,ltrlz becomes a single-valued

(holomorphic Abelian) differential on s. Applying the argument below to this

differential on S, we have the assertion for the general case.When rl.'Llz has a single-valued global branch, say d, which is an Abelian

differential, we define the geodesic flow {.F1 | t € R} on S with respect to the"metric"

ldl. To explain the construction, we always take

as a ry'-coordinate around p which is not a zero of ,!.We continue the inverse

mapping tr;l along segments on R in both directions as far as possible' Then

we get a locally biholomorphic mapping, which is denoted by the same notation

V;l , ol a domain containing an open interval 1o = (r1,ur)--of.R into .9, where-'oo ( u1 1u2 ( oo. we set f lo =vor(I). This flo is called the |-horizontal

lize passing through p. It is also called a horizonlal trajectory of r/. (See Fig.

5 .2 . )Note that, when we trace along flo in one direction, either flo ends at azero

of r!, or lc.rl tends to oo.In particular, restricting ty'-distance on Hp, we can identify.[/o either with a

circle or with a subinterval, say /o of R, preserving orientation and length. Let

u = irrr(q) = loo

t

1eB arrr 'lfil "l lradsa.r WIm oraz€ere spq u - s - or ecurs'u x et uo uorlDunJ alq€rnseeu e s (?'D'd)y ueqa

'u I ?'U ) b'((b),ul'6)y = (t'b,6)y

1es a.n '1xag'S'uo

.(lueruala seJ€,, eqt s€ tpop srql esn e.rrr,uo a.raq uolg .ltil ((crJleru,, aqlo1 lradsar ql1,rn Sura.raserd-eare q t4, ''a.l ,U

Jo X ?esqns elqernseeu fraaa ro;

xrr (x)urrrpop ll -"pop

ll JJ JJ

leql aes o1 .,(sea sr 1r '.re1nar1red u1 'saleurproo,

-@ o1 lcadser qlr.&r I i{q ,,uor1e1sueJ} I"luozrroq 1a11e.red,, eq} sluesa.rdar qcrqrrr'6

;o Surdd€ur-Jles elq"rns€aru arrrlcaftq e sl ?d fraaa 1eq1 ees uec elrr uer{I

'U)d 'U)?'(t)dV=@)tg

3ur11as ,tq rg augep eAyA - S -

t;i las pue 'arogeq se eq gr p1 .lfil (clr?aru),aql o1 lradsal qll^{ S uo {U ) ll tdl ^rog crsapoe3 aq1 augap yleqs e,r,r, 'a,ro51

'S uo ldl ((crrleru,,aql pue U uo )rrlatu u€eprlcng aqt of lcadsar qlr^\ rrrler.uosr f11eco1 sr qrlq&l

'd = (0)ol 'dH *- E,dl

Eurddeu€ ur€lqo a,u 'ul

,!n = d7 3ur11as 'fl - S 3 d f.rala roJ snqJ .lr7tl ,,crr1eur,, eq1

o1 lcadser ql-r.{ oraz "ere s€rl 3' }eqt ees uB) e1r{ ,sotrez Jo Jaqrunu elrug e fluo

seq qt aeurg 'U Jo Ie raturqns radord e sl dI leql qcns S ) d Jo tes eql eq A

(g p ol:,z aldurrs p rpau saurl'z'9'tlJ pluozrroq-p)

\\

\

l--,l, r,-

\ | /r \ tt/r\l lluaroeqJ ssauanbrun s(rellnruqf,ral Jo Joord

'g.g tlI

t44 5. Teichmiller Spaces

(5.8), = I"( l l ,^r, t , t)dodr) dt

= l: "(l 1,,,',)(g' c) a"a') at

= ilrls(Lr)ledodr.

(5.e)

desiredtr

Notes

For the reduced Teichmiiller spaces, see for instance Earle [57].When f = {id}, we denote by "(1) the corresponding Teichmiiller space

?(f), and call it the aniuersal Teichrniiller spoce. see Lehto [A-68], Chapter-III.

The fact that "(1) is contractible was firstly shown in Earle and Eells [62]. In

Douady and Earle [53], it is proved that "(f) is also contractible for every f.

For investigations on the Teichmiiller metric from the differential-geometric

viewpoint, see Kravetz [128] and o'Byrne [169]. The Teichmiiller metric on

?, is not smooth. See Earle and Kra [65], Royden [184], and Gardiner [A-34]'

$9.4. Moreover, ?o does not have non-positive "curvature" with respect to the

Teichmiiller metric, as is proved in Ma.sur [142]. See also Theorem 6.21 in Chap-

ter 6.Teichmiiller's theorem gives another compactification of the Teichmiiller

space ?e, which is called Teichmiiller's compactification of 4. This is differ-

ent from Thurston's one defined in chapter 3. see Kerckhoff [111] and Masur

lr46l.- .itr" original *proof' of Teichmiiller's theorem is found in Teichmiiller [A-106]. The proof in this chapter follows that in Bers [23]. We also refer to Abikoff

= l:"(l l,^r, d dodr) dt = 2L I l,^ro,

q) dodr

for every positive .t.On the other hand, since g is quasiconformal, and hence is ACL, we see that

Fubini's theorem gives

, = I I,(l _"ur,t,i at) aoar

Here, we set .Lo - lr([-t,.L]), and hence lLolq =2L.Finally, applying Lemma 5.17, we conclude from (5.8) and (5.9) that

2L [ [ \(s,q)dodr>(21 - t t1 [ [ a"a,-J J S J J S

Divide both sides by 2L, and let ,L tend to oo' Then we obtain theinequality (5.6).

'[Orz] '[Oea] qcea1 pue ,[666] rqcn8ru€J ,[67I] ,[gtt] ,[rtr] ,[qrr] ,[Wt]

rns"trAtr '[gtt] "I[*S pue msstr l 'goqqo.ray ,[lg] pr"qq.,H pue ,(penoq ol reJarosp a \'[ZO1-V] Ieqarts pue'[Z]-V] sur{uaf ,[lS-V] raurpreC ol reJer a^{,g$ur se sl€rlueJagrp )rlerpenb crqdrouroloq Jo setpnls lecr.rlauroa3 .rog ,flpurg

seloN slr pu" y xpuaddy ur pelelsare sef,eJrns uueruorg praua3 yo suorleuroJep FuJoJuotrsenb uo scrdol aurog

'[291] ue4eg pue'[08] .raurp.reg'[62]ue{es pue uueurlqeJ '[Og-V]

(p{qsnry alr) e,tr ,s3urddeur leruroJuocrs€nb 1eure.r1

-xe uo suorle3rlsa,rur raqlo sy'[016] Iaqerls pu€'[0/I] a{"tqo,[gg1] ue11nrycryaes 'l"uerlxa dgressacau lou $ e?eJrns Eurrarloc e o1 Surddeu l"urroJuocrsenbIsuerlxe u€ Jo lJrl e 'alourrarlllng 's3urdderu

IsruroJuocrsenb leurerlxa Jo sseu-anbrun uo s?lnser ureluoc [961] sareqles pue,[6] ,I"l"U pue treurfell ,os1y'[gg1]

1aqa.r1s pue qcleg eldurexe roJ aas 'l€tuer]xa ,(lanbrun aq ol lou paeu lr'leurarlxa sr Surddeur ra[nuqrral FurroJ e Jr ua^g 'letuerlxa fpressacau ]ou artss8urdderu rellnurqrreJ l"ruroJ aIq.,rr 'sSurddeur

leur.ro;uocrsenb leuerlxa flanbun

Ilrls a.rc s3urddeur rellnuqcrel 'ece;.rns uueuarg lelauaS € Jo es?r aql uI'qrstep e.rour roy [gg-y]

otqerl pus [pg-y] .reurpreC ees '([I IA] leqarlg

';r) auo tuercgns e oEe sr uozlrpuo?spollnang slql 1eq? pa$oqs leqarts pue qrrsg ,la,roero141 .uaroeq? sseuenbruns(rellnur{f,ral alord o1 ruaroeq} s(uollrru"H slql asn uec a11t .Fuerlxe aq olSurddeur leruroJuof,rs?nb e .roy uorlrpuo? fressaceu e srsfleue IsuorlounJ Sursn ,tqpe,lord [69] uollueg '(asuas eruos ur purrunu) Itsuerlxa sr Surddeur JellnuqcraJe Jo luel?lgaor nueJllag eql leq] sa]€ls rueroeql ssauanbrun s(rell3ruqtlal .[I-y]

ItIseloN

Chapter 6

Complex Analytic Theory of TeichmiillerSpaces

We introduce a natural complex manifold structure of the Teichmiiller space

"(R) of a closed Riemann surface R of genus C(> 2), which is realized as a

bounded domain in C3s-e. Furthermore, we prove that the Teichmiiller mod-

ular group Mod(R) acts properly discontinuously as a group of biholomorphicautomorphisms of ?(,R).

In this chapter, unless otherwise stated, we assume that l- is a Fuchsian

model of a closed Riemann surface of genus S (Z 2) and that each of 0, 1, and

oo is f ixed by an element in l '- { i,d} (cf. $1.2 of Chapter 5).

In Section L, following the idea due to Bers, by using Schwarzian derivatives,

we prove that the Teichmiiller space ?(.1-) is realized as a bounded domain Tn!)

in the space A2(H. lf) of holomorphic quadratic differentials on the Riemannsurface H*/f , where 11* is the lower half-plane. The Riemann-Roch theoremshows that A2(H. lf) is a complex (39 - 3)-dimensional vector space. Hence

TaQ) is regarded as a bounded domain in Csg-s. Identifying "(l-) with Tn(f),

we see that T(f ) has a complex manifold structure of dimension 3s - 3.

In Section 2, we show that this complex structure of "(f ) is independent of

,| , that is, ?(f) is biholomorphically equivalent to T(ft) for another Fuchsianmodel f' ol a closed Riemann surface of genus g.

It is verified in Section 3 that the Teichmiiller modulat group Mod(f) off acts properly discontinuously as a group of biholomorphic automorphisms of

"(f). Thus we conclude that the moduli space Mn =T(l)lMod(f) has a nor-

mal complex analytic space structure of dimension 3g - 3. In Section 4, we shall

explain Royden's theorem which asserts that every biholomorphic automorphismof "(f) is induced by an element of Mod(f).

Finally, in Section 5, we give a brief exposition of the Thurston-Bers theory

on the classification of Teichmiiller modular transformations.

,tq uarrrS e * ? :/ ursrqdrouroeuoq e el€q elrr uaqt '11 uo ,rn - ,1m 11 'too.r6

' *H uo nn = nn (rr)'U uo nn = tn (\)

:luel-oamba a.r,o 6utmo11ot eW'r(J'H)g > n'rl s7uau,a1a omy fi.uo Jotr 'T'g BrrruraT

'fle,rtlcedser 'n.t l!,U pue nJ

ld H o1 crqd.rouroloqlq arc

*U Jo U a3eurr .ror.rrur eql pue S ?erll epnlf,uoc errr 'uotlezrurroJlun snoeuellnruls

,s.reg ,tq ueqtr '/ Jo luelcgeoc rurerlleg aql 'trl - r/ tas

'S o1 *2f ;o / Surddeur

IeuroJuocls?nb e ar1e1 Pu€ 'U;o

*gt e3eurt Jorrnu eql Jo J lepour u€IsqcnJ e rlctd'1ceg u1 ',S pue g flsnoeuellnurls sazlurroJlun qclq^r /.7 dno.r3 uetsqcng-rsenb epug e r 'f snuaS;o S pue U sec€Jrns uuetuelg pesolc o,ra.1 due ro; 're1ncr1.red u1

'([76] srag aes) uotToztrulolrun snoaunlputs.sreBr Pelle? q slr{J,'d.7 dnor3 u€rsq?ng-Isenb e13urs e ,tq ,(lsnoeuellnuls pezluroJlun er€ +Ur pu€ ttg

saceJrns uuerualu orrrl'f1e.,lr1cedset'd1flg pue nJ/nH,{q peluesarder er€ *Urpue ttg saceJrns uueualg o,lrl eculs'lltt = g;o e3eun rorrflu et{t sl *2f eraqlrr'Ulln oI Jl *H = *U Jo Surddeur crqdrouroloqlq e pu€ h/nn = dA o+ JIH= g' Jo Surddeur leuJoJuoclsenb e sacnpur trn, Surddetu leuJoJuoctsenb eq;

'in put rtg qloQ uo slurod

paxgou seq {pl} -nJ Jo }uatuala f.rarra 1eq1 eloN'paxg C uI elrnf, pasol) eldurts

palcerrp € sa^eel q)lqrtr (c'z)lsa 3o dno.rqns alarcslP e 'uorlrugap fq'st dno.t0uDrsqcnl-rsnnD e 'a.re11 'sdnor3 uelsqcnJ-Isenb go eldurexa lecrdfl e sr d.7 dnorte q?ns'GH)n^ -

ltl pt* (U)'* = "-t1 qtoq uo {lsnonurluo)slp.{1.redo.Id slceqcrq,n'(3)lnv Jo {J ) Ll (L)'X} = nJ dnor8qns € eleq a^\ snql '(q)t"V

1"

lueruele ue ''e'r'uorleurro;su€J1 snrqotr^tr e sr (1")/X leql eas eu'g raldeqCJo t'I$ul ?eql o1 Suruosear rBIIruIs fq 'r-(/rn)o Lodm = (1,)/X 3ur11nd 'J f ,L fue rog

_"'m

.{q 1r alouep a,u '(gg't uorlrsodo.r4) H lo ,tn Eurddeur ob-r/ lecruouec eq} uorJ

Surddeu IsurroJuotlsenb srql qsrn3utlstp ol repro u1 'flerrtlcadse.r 'pexg oo pue '1

'6 saaeal pue '/ uorlel€lrp xelduroc eql seq qrlq.tr ? go Surddeur leruro;uocrsenbe ''a'l '?

3o Surddeur cb-r/ lecruouec e dlanbrun slsrxa areql '08'? ruaroeql tuord

'H-c.)z

H>zles e1(

'J roJ Ff uo r/ luercgaor rruerlleg e ''e'l'r(l'n)S 3 r/ lueuele uarr,rE e rog'*H

aueld-;1eq re^\ol eql uo leuroJuoc er€ q?rq^\ C a.raqds uuetuelU eql;o s3urdderu

IeruroJuocrs€nb fq (.7)g aceds rallnurqclel eql luasardar lleqs elri 's.reg Surrrrollog

uol+BzrruJolrrlf snoauBllntllrs'T'I'9

arBds rallntuqrraJ,

pue Eurppaqtug (srag 'I'g

,(,)2\ = ?)d

Jo arnlrnrls xalduoC aql

LVISurppaqurg ,srag'I'9

148 6. Complex Analytic Theory of Teichmiiller Spaces

z € H

z e I l * U A .

Since (tot')-rotul is quasiconformal on C, we see fhat f is ACL on C. Thus, bythe analytic definition A of quasiconformal mappings ([1.1 of Chapter 4), / isquasiconformal. Hence, g = wpof o(wr)-1 is a L-qc mapping on C, i.e., a Mobiustransformation. Since g leaves each of 0, 1, and oo fixed, g must be the identity.Therefore, we have up = u, on I1*.

Conversely, if wu - IDv on I/*, then utt = It)y on Iy'* U R. Thus we obtaina quasiconformal mapping h - wpo(wp)-Low,o(w')-L: H - H.By the sameargument as before, it follows that h must be the identity, which means thatwF = w' onR..

Now, for two elements p,v € B(H,f)1, wu and'wv are said to be equiaalentif wu- w, on H*. Denote by [ror] the equivalence class of wrfor every elementp e B(H ,.1-)r. Let fB€) be the set of these equivalence classes [tor]. Lemma 6.1shows that the correspondence l*,) * [tor] is a bijection of "(l- ) to "p (f ) . Thetopology of TBQ) is induced from that of "(f) under this correspondence. Inother words, this correspondence gives a homeomorphism of 7(f) onto fBQ).In this way, we can identify fpQ) with "(f) as topological spaces. We also callfp!) the Teichmiiller space of l.

Let B be a mapping of B(H,f)r onto TBQ) given by 00t) = [or]. Then bythe definition of topology olTB(f), we immediately obtain the following.

Proposition 6.2. The mapping B: B(H,f)1 * fBQ) is a continuous surjec-t ion.

One merit of the Teichmiiller space TB(f) introduced by Bers is the applica-bility of the theory of univalent functions, i.e., conformal mappings of H* .

6.L.2. Schwarzian Derivative

A quasiconformal mappin E u p 6 defined in $ 1.1 is conformal on the lower half-plane 11*.

Now, assume that wu is a Mcibius transformation. Since uru leaves each of 0, 1,and oo fixed, tuu must be the identity. Thus we have [tou] = [rd] in fBQ).It maybe considered that the diflerence between [tou] and liQ in fpQ) is indicated bythe difference of the conformal mapping wu on H* from Mobius transformations.

To measure the difference of a conformal mapping on I/* from a Mobiustransformation, we shall find a differential equation which all Mcibius trans-formations satisfy. Let 7Q)

- (az +b)/(cz * d) be a Mijbius transformation,where c, b,c,d e C and ad - bc = 1. Take derivatives of 7 to eliminate a,b, c, and d. Since j 'Q) = @z + d)-2 and l '(z)

- -2c(cz + d)-", we obtain

l 'G)/t"Q) = -z12 - dl2c. Thus we have (lf (7" 1t')) ' = -1/2. Consequently,we get

f ( " ) = { ( " ) - t " ' ' ( " ) '

pruroJuor e Surugap uer{J'*Il uo ndt - ndt leql arunsse'f1as.ra,ruo3',I/ uondt = f,dt 1eq1 sa11dur1 {rpla'.II uo not - drn ueql '(,1)d,t ul ["m] = [dt]lI

'(6'9) elnur.roJ e eq a.tr smlJ'*I/ uo {r'n^} = "(z),L{(z)L'dm } 1aB a,u 8'9"urrrutrrerl Aq(rt6ortL - Lodm Jo e^rle^rrap uerzrs^{qcs eq1 3ur:p;'t(;(g)g ut st

r/ asnecaq 'uotleurro3.suer? sntqoni e sr ,-(/rn)oLodm = dl uaql 'J > L 11 'loo.t4

'*H uo nd - 'td) fi fi1uo puo

fi Q)gl,q [,n] = [n.]'t(J'H)g ),7'tt etu?uep omy f,uo.tot'taaoa.to141'J

/ ,H acottns

uuoulery D uo lorlueta[ry c4o.tponb ctyd.toruopy o co papto|a.t st 7t puo ',1 o7

Tcadsat, q??n *H uo V- Tq|nn lo ut.tot ctyrltou.toTto atyiLtoruopy o sN ndt 'fipu.to7J

(z.g) .*H)z ,(t)nd="(z),L((z)L)d6

ueql 'J ) L lI 'V'g BturuaT

'3urmo11o; aql e^er{ e.&r ueqtr,

'*H)z '{z'dm}=(z)ddt

las e,/rr'r(J'H)g 3 r/ r(re.rltqre rog

acudg

ra[nurqcral Jo arnl"nrlg xeldurog aql PrrB Eurppaqrug 6srag 'g'I'9

tr 'uolleturoJ

-suerl snrqontr " q / feqt apnlcuoc aan 'uorlenba FlluaraJlp slql 3ul^los 'O uo

o= r{,((r),!sq)}f -,,((r),!sor) - {''!)

leq? lno surnl lr'O uo 0 - {r'I};r'flasreluoC

'O uo 0 = {z '/} sagsles / uotpurlo;suerl snlqotr{ € 13ql uees

itpearle e erl e1yuorlress? tsrg aql sn sarrrS uolleln?Fc preruro;lq3le r1s 'rg 'loo.r'4

'O uo 0= {t't}19 fi1uo puv fi uotyout.rotsuorl sn qory o s? CI {o furddout' Tortt"t'otuoc o 'teaoa.r,o141

(r'g)'Q) z '{r'l}+ "(z),t.{Q)t

'f } = { z'to6}

u?tll'fi1aat1eailsa.t '(O)l pu, q to s|utildotu lotu.totuoc atp 6 puo I It

't'g BtutuaT

(94\s-!',)'Ii={,'!},\(r),J ) t (z),,,!

,(q / lo {t'l } aaqoarreP uotzrDnqrsaqt auuep e^{ 'C uI ururuop e uo 3[ Surddeu l€trrroJuoc frerltqre ue rod

( (z),L \ T, (r),L,-.\I;'Z)E-GlJ,

6'ISuppaqurg .srag 'I'9

150 6. Complex Analytic Theory of Teichmtller Spaces

mapping F: wu(H*) - u,(H*) by ,F = w,o(uu)-L, again by Lemma 6.3 we seethat

g , ( z ) = { F o w p t r } = { F , w u Q ) } ' . ' u Q ) ' + p p / )

on 11*. By the assumption that g u - g v on H *, we have { F, z }

- 0 on uu(I{' ).Thus .t' must be a Miibius tra.nsformation. Since f leaves each of 0. 1. and oofixed, we see that F is the identity. Consequently, uu = '.i)v on f1*, that is,

l.ul= [u'"] in TBQ).

Let A2(H*, f) be the complex vector space of holomorphic automorphicforms of weight -4 on -I1* with respect to f . Since it is identified with the vectorspace A2(H- / t) of holomorphic quadratic differentials on H* f | , the Riemann-Roch theorem shows that A2(H*, f) is a (3S - 3)-dimensional complex vectorspace.

Now, define a mapping B of :tBQ) into A2(H*,f) by B(lwrl) = pu, wheregp= {up,z} , the Schwarz ian der ivat iveof wu on f I * . Then, by Lemma6.4 th is6 is well-defined and injective, and that is called Bers'embedding.The mappingiD: B(H ,I)t ---* Az(H* , f ) given bV @(p) = 8"0(p) is called Bers' projection.

In $2.2 of the previous chapter, Az(H* , f) was considered as a complexBanach space with trr-norm. In this chapter, in connection with the next sub-section $1.4, we introduce on A2(H* ,f) the hyperbolic -L--norm by using thePoincar6 metric dss.z = ldzl2/(Imz)z on f1* as follows. By formula (6.2)and the inva,riance of the Poincar6 metric under P.9.t(2,R), every elementp e A 2 ( H . , f ) s a t i s f i e s

(Im1Q\2le(.r(")) l = (Imz)zleQ)1, z e H*, 1 e r.

Thus, (Im r)2lp(r)l is regarded as a function on .R* = H*ll. The hyperbolicL* -norm of 9 in Az(H* ,l-) is defined by

l lpl l* =,s.S.(Im z)'� leQ)|

Here, note that the supremum suffices to be taken over, not the whole If*, butonly a fundamental domain in 11* for .l-. In our case, .R* being compact, wecan pick a relatively compact subset in f1* as such a domain (see Example 5 in

$4.2 of Chapter 2). Therefor", l lpl l- is f inite for any p € A2(H*, l-), and henceAz(H*,f) becomes a complex Banach space with this norm. Throughout thischapter, we assume that A2(H*,1-) is equipped with this norm.

Proposi t ion 6.5. Both Bers ' pro ject ion Q: B(H, f )1 * Az(H*, f ) and, Bers 'embedding B: TBQ) - A2(H* , f ) are conlinuous.

Prool. Note that { p"}T=t converges to <p in Az(H*,f) if and only if {p"}f;=rconverges to g uniformly on compact sets on f/*. Hence, Proposition 4.36 impliesthat @ is continuous. By the definition of topology of Tp(f), it follows that B is

tr

also continuous.

I=l

7r - u7_r7l"ql" 3N

1eq1 seqdur qcrqrrl

/t=\'o-

l,r-'"1"s1"3 - "tl o - "v\o/

al"q eA\ ' gz?J = m rc1 mf ".t - qr, 3ur1ou 'snq;

"3f'@),rp@)t"l *=", -J

I

fq ua.,rt3 sl iC ,tq pepunorrns uretuop pepunoq eql Jo 'y

eere aql 1eq1 sarldurr elnuroJ s(ueerD ueqJ 'd repun { .r = lnl I C > .l = "C

elcr.nr eqt yo a3eun aql aq ',C 1el

'I < .r ,{.rerlrqre ro;'1ce; uI'I t |tql ,r"qt

(e'g)' +t*t*oq*m=(rn)g=)

fq ua,rr8 q rf JI 1er{} aes lleqs e A 'y

{slp lrun aq} Jo rorre?xeeql 'y -

? -

*y ereqa '*V uo Jr uorlcunJ luale^run € Jeprsuof, aiu '1srrg 3foo"r4,

'! 'H)z

tr; l{ z'{}1"(zw1) dns - *ll{"'/}ll

fr,Tqonbaue aylsa{sr,7ns *H uo uotTaunt lualDarun fitaag (snutx puB r.reqag) 'Z'g BrrruraT

'sneJy pus rJ€qeN ol enp sr rIJrrIA\ '*Il uo suollounJ

luale^Iun ro; ,(lqenbaul ue Jo ecuanbesuoe el€rpeuurr ue sr rueJoeql slt{I

'1fgsntpo"r puD 0r?lun q?gn (J'*H)GVu? nvq uado aql u, peurDlu@ s? (ilal acoils .tapnu,?pl4 eUJ 'g'g uraroaqJ

'(J'*H)zV ur ureurop pepunoq e sl (J)sJ l€rl1 slrroqs uraroaql 3urno11o; eq;

Q)slJo ssaupapunog .t.I.g

('9'6$ aas 'oqy 'asec l€uorsueurp-auo eql JoJ leql ol relrturs sr ploJrrr€ur

xelduroc Ieuorsueurp raq3rq s Jo uorlrugep eqJ) 'J/H = Ar eraq/!\ 'sp1o;rueu

xelduroc Ieuorsuaurp-(g - 0g) se peraprsuoc osle are (g),, p"n '(l)gJ'(I)t

saceds rellnurqcral aql '(J)aJ qlr^r uollecgrluepr rapun '1 1o acods relpu.tqcteJ

aql palleo osle sr (l)sJ slql '(J'*H)zV Jo ernlcnrls ploJrueur xalduoc aq1

slrraqur (l)al,'aceds rol?el xaldtuoc leuorsueurp-(g - 0g) e sl (J ',p.)zy ecurg'uruqdrouroeuoq e q (J)sJ -- Q)d,l:g prre'(J'*H)zV ur ureruop e sr(J'*H)zV * (t)d,f,:g uorlcafur snonurluoc eqlJo (J)sJ a3erur eql1eq1se11dur1sureuopJo ecuerJsAur uo rueroeql s(rea{norg snql'(.i,)ag sr os pue'g-ogtl ol

crqd.rouroauroq sl (J)J areds rallnurqclel aql '91'g ue.roeql ur pelels sV

I9ISurppaqurg,sra6l 'I'9

152 6. Complex Analytic Theory of Teichmriller Spaces

for any positive integer N. Letting r * 1 and then letting N'--+ oo, we obtainthe inequality

i " 'u l , s 1.n = L

This is the content of the so-called Bieberbach's arta lheorem. In particular, wehave lDll S 1.

Differentiating the series in (6.3) term by term, we obtain

{4 , } = -#0 , * i# , weA* - { * } .

Hence, we get

.14$l . , ' { r , . } l= 6 lar l< 6.

Now, let / be an a,rbitrary univalent function on I/*. For a given point zo =ro * iao € H* , first suppose that f (r.) * oo. Taking a Mcibius transformationT: H* ---+ ^4* defined by "(z) = (z -Z;)lQ - zo), we put

F(u)=r@w*, w€A*.

Then F is a univalent function on A*, and has an expansion as (6.3). Flomformula (6.1), we have { f , r} = {F,T(z) l .T'( t )" on I1*. Thus not ing T(t") -

oo and T'(r)'= -4v|fQ)alQ - %)n, we conclude that

l { f , r , } l =, l l l } " l { r , rp1}. r ,Q)r l

= .rgg l*n{ r,, }l .,!g" & S #"

Next, suppose that f (r") = oo. Then by the relat ion { f , tol - { I l f , r"}and the above argument, we see again that l{ f,zo}l t Sl!yl). Since zo isarbitrary, we complete the proof of Lemma 6.7. o

6.2. Invariance of Complex Structure of TeichmiillerSpace

Let us prove that the complex structure of "(f) which is introduced in thepreceding section is independent of the choice of the Fbchsian model ,l- of a

closed Riemann surface of genus c(>2).

(g'g)1-\bru-zultt

eleq eir\'*I1 uo g= ,(lltrlt, - z&l&) eculs

(q'g)

uorlrpuor uorl"zrlerurou aq1 f;st1es zb, pue It t"q1 etunss€ a,t. 'ata11

(r'g)'g=bdt9q,,tt

uorlenba lertueragrp freurpro repro-puo)es aq1 ;o z& pue tlr suorlnlos arqd.rouroloq luepuadapur fl.reauq a4e1 'd e,rr1-e^rrap uerzr€^rqrs qll,lr *I/ uo uorlcunJ luele^run 3 Irnrlsuoc o+ repro u1 '{oo.r,4

'dt = (l"dnl)gsa{n1os d> uto.{ p?prul.suo? 611

1o4uau$rp Nravrqeg ?Nuoruroy ?Ul '2,/I > -lldll

qlr,n (J'*H)zV 3 dt Tuauale fiuo.r,og (tU"rf4, pue sroJIqV) 'O'S tuaroaqtr,

sroJlr{y or anp ruaroaq} Bur,u.o11oy aqr Jo acuanbesuoc "r"ro"**,'|lJl ttt'",ttt 1""

'p! = ttrog \?pm ,utddout ctyd.tou-to1or1 o''a'!'A<-

2:glo esreau, f16ur o s! /)<- Ai4puD '(l)dt ut Tutod asoq ay1/�o

pooty,oqq|nu uedo uo s! (A)4 - 2

'uoqon7ts |utpaca.td ell repun '8'g uraroaql

'6'9 uorlrsodoJd ruo+ snonur?uoc sr qtlq^r

'[^n*) - (d),ntq Q)d,t +- A i 4

Surddeur € euuap e,tl snql ' A ) o\ itra,ra ro3 .1, o1 laadsar qll^{ }uenlgaor nuerllege ''e'l 'r(t'n)A o1 sEuolaq drl ueqa '(l'*n)zv ur urSr.ro eql Jo pooq.roqq3reun'{Z/t > -lldll | (t'.n)"V ) 6} = A1e.I 'crrteru ?r€f,urod eq} fq pecnpulrolerado ruerlleg-er"1de1 aq1 o1 lcedsa.r qlr^a urroJ (t 't-) cruorureq € sr lI ef,uls'p4uataSrp ,urDrlleg ?ruouuDq e pell"c osle st 11 'o1 uror; pelsnrlsuo, lo4uetal-lrp tutotTlag .sreg eq! pellsc sr "/.HI'(*t'n)A 3 drl luaruele ue ulslqo aa{

'H ) z '(z)dt"(zwl6- =Q)drt

3ur11es snql 'J/.F/ uo lerluareJrp rurerllag e s zp/zp(z)dtr(z ul = Itp/"r!Q)fi_ueqJ'II uo rrrleru er"f,urod erll eq ,(zu4)/"lzpl = Itp p.I'H 3 z fue .roy

Q)a = Q)fr fq paugap (l'tt)"V ) ql luatuala ue 1aB "^ '(J ' *H)zV 3 al luaurela

fre.rlrq.re u€ roJ 's,raolloJ * (,1 ';J)g 3 drl luaruala ue qlra (tr' .11)cy ) dtluauela qcsa elercosse e,ra '(;)ag ur urSr.ro eqt Jo pooq.roqqSreu e ur (,t)s,l* (l)d,l:g Eurppaqure (sreg Jo asralur crqdrouroloq e f1t1c11dxa lcnrlsuo? oJ

tmppaqurg 6srag Jo asralul IBcoT 'l'Z'g

aredg rallnurql-ral Jo ern?f,nrls xaldurog Jo af,u"rr"^ul 'Z'g

't = (t-)zu = (l-) I& \O = Q-)It" = (p-)rh )

t9I

t54 6. Complex Analytic Theory of Teichmiiller Spaces

on ff*. Set /(z) = ,nQ)lrtz(z) for any t e H^*.From (6.6), we see that / isa locally biholomorphic mapping of 11* into C. A straightforward calculationg i v e s { f , z } = 9 o n H * .

Now, we put

( o . { )

Then ,F' is a real-analytic mapping of /1 into e , b"""n." its numerator anddenominator do not vanish simultaneously from (6.6). Bv a simple computation,we see that Fsf F, -

Hv on I1. Since llprll- ( 1, the Jacobian of P is positive

on 11, and hence F is locally diffeomorphic on 11.Next, we set

: . \ | F ( r ) ' z € HI \ z ) = l / ( r ) , z € H * .

We need to prove that f exte^nds to a quasiconformal mapping of e onto itself

in such a way that upn = So/ for some Mijbius transformation S, which implies

t h a t { u p , , z } = p .For this purpose, first suppose that g is holomorphic in a neighborhood of

,F1. U fr. in 0 and lp(t)l = O(lrl-n) as z ---+ oo. Then ?r and qz are defined on

a neighborhood of the real axis \, and so are / and F. Since f = F on R,we obtain a continuous mapping f of C into C by putting f = f on R. Thisextended mapping f is locally homeomorphic on a neighborhood of R.

In fact, for any point z on R, choosing a small disk D with center z, we see

that both f : D--- /(D) and F: D---+ F(D) are homeomorphic, and f = F on

DnR. Since both / and F are orientation-preserving, f(Dn I1*) and F(DnH)

do not intersect, which implies that / is injective on D. It is easy to see that fis an open mapping, and hence f^: D --- /(D) is a homeomorphism.

Moreover, it is proved that f also extends to a local homeomorphism on e

as follows. Since l9(z)l = O(lzl-a) as z'--+ oo, r/1 and \z ate expanded near oo

in the formqte) = a1z * bt + O(lzl-r),

nz(z) = azz * bz + O(lzl-L),

where c162- a2b1 = 1. Hence, f = rylnz is a univalent function in a neighborhoodof oo, and .f(*) = at/az. On the other hand, we have

F(z) =a t z * b t + O ( l z l - 1 )

(z * oo).a 2 z ! b 2 + O ( l z l - t )

Thus F is an orientation-preserving diffeomorphism on a neighborhood of oo

with ,F(oo) - ar/az. Therefore, putting l(o") = at/az, we see that / is a Iocal

homeomorphism of 0 into itself. Then Lemma 4.28 implies that i: e --* e it u

homeomorphism.Applying Painlev6's theorem t?.r,o(.f)-1, we see that there exists a Mobius

transformation S with wp, = Sof .

To remove the hypothesis that g is holomorphic on R with zero of order at

least 4 at oo, we pick up a Mcibius tra.nsformation fl. given by

F(z)=ff i , z€H'

uorlenba rtuerlleg aq1 3o (1'9) ut.to;

eql ul uoltnlos e pug ol poqlau cll$rnaq e ureldxa ol e{ll plno^^ eM'6 qrvueq

'lcedtuoc sl JIH leqt srsaqloddq eql asn lou plp

a,r,r;oord aql q ecurs'.-;' dnor3 uetsqcng.,(ue roJ sploq 6'9 tueloaqJ,'[ slrDuev

! '6'9 ueroeqJ ;o Soord eq1 salelduro? slqJ

' *H uo 6 -

{r'los} = {z'6rim) = (l^'^l)g

urelqo eal 'ra,roe.roy41 '3 uo 'drnor-5 Surddeu leturo;uoctsenb€ ol spuelxa / teqt seqdtur qclq,\{'lI - C uo ^amor-S = ;l'pue'(q)nV u1

S - "S ler{l ees arr,r 'relncryed uI 'U - C Jo slesqns lcedruoc uo (orrleur lect.raqds

eql o1 lradsar qp.,u) fpro;run 3l <- $ 'uorlcnrlsuoc fq snqa '.{1a,rr1cedsa.r '?l.l

pu€ Il, oI *H Jo slesqns lcedruoc uo ,{pu.ro;tun a3ra,ruoc ud t'o1 (7'9) uotlenbe

I€rluereJ-rp "ql p u'zb pu€ u'rl, suollnlos pezllerurou eq? 'pueq raqlo eql uO'C

Jo slasqns lceduroc uo flurro;tun ^dot *- "dotr

'.{lluanbesuoC'C Jo slasqns lcedtuoc uo fpruo;run P! * ud teq} ees am'2'9

eurrreT ;o goo.rd eql ul se luarunSre atutss eql fq 'ecua11 'U - 3 Jo slesqns

lcedtuoo uo ,{prroSrun 0 * "6rl Wql sarldtur gl't uoltlsodor4 ur (8't) '.t"ql

r > (?llalF + I)/-lldllt ; -ll"u'lll

urctqo a { snql 'z{(-lldllA - I)/(*lldllZ + I)} J uotlelepp l€ulrxeure seq 'f leql ees e/tr 0I'7 ueroeqJ uror.;: 'r-(^ilm)o"dm = u6 3ur11e1 ug

'u5 uotleu.roJsuerl snlqgl tr etuos roJ u{ous = "drn 1eq1 qcns

Jleslr oluo e J" "! Surddeur ler[roJuocrsenb e sernpord tol 'uant3 uollrnr]suoc

aqt,{Q'ecue11 't > -llallZ S

-ll"dllZ = -ll"t/ll 1eB er't t"6rl = 'rl 3ur11nd'rrlo1q

'9 t -llrll ; TL

l(@ "t)al. 11 z; 2 ur1; f,riJ ;

l(r)"otlr1rql f,,i." = -ll'dll

. ,.\, ,z((t)";*I) =eQutl)*H )' -fFtt-'

r1eE a,r.r, '1'g uotltsodor4

de'*11 ) Gn)"t Eur1otr1 'oo le ? ls€el 1€ reProJo otaze seq Pue'(p'),l,Z.loaprslno paugep uollcunJ crqdrouoloq e sr ud> teql aes el.l.'"(z)ia' (Q)"a)d = "6

3ur11ag 'oo F u s€ pl o1 C Jo slesqns lceduroc uo ,tpu.ro;run sa3re,ruoc "J pue ' H

Jo ur€uropqns lcedruoc flarrtlelar e sr (g)1lj ueql'u.re3alut arr.tlrsod,'{ue ro;

u7 -t zl= (z)".L

?-zuz

acedg rallnurq)-reJ ]o arnlf,nrls xalduroS Jo af,u"rr"^ul 'Z'9

e^eq e.!\ snql

991

156 6. Complex Analytic Theory of Teichmriller Spaces

I ,w, =

)(z _ z)2 ee)w,

on fI for any p € Az(H*,1-) with llpll- < 1./2.This is due to Shigeru Furuya,and the authors learned it from K6ta^ro Oikawa.

setting z - y and z = sin the above Beltrami equation, we have apartialdifferential equation

Lw , = i @ - x ) z e ( o ) w u .

Denote by w(x,U) = C with an arbitrary constant C a general solution of Ric-cati's differential equation

Iu ' = - | @ - r ) 2 p ( r ) .

Then this u gives a solution of the above partial differential equation.Pu t t i ng u=U- o , wege t

u ' + r It - - rv@) '

Thus, settingu= -af a/, we obtain the second-order differential equation

Iu, , = _)v@)a.

Take linearly independent solutions [1 and rp of this equation. Then we see that

- . . - T t + ( Y - ' ) n ' r- - ffi|;--;rt'and hence we obtain F' in (6.7).

Corollary. For eaery g e V, therv ezists an element p e B(H, f)1 such thatw, is real-analytic on H and B-r(p) = lup).

Mortoaer, eaery point [S,f] of the Teichmiller spoce f@) "f R - H/f isrepresentedbyareal -analyt icquasiconformalmappings of RloS, i .e . , [S, . f ] =

lS,sl in r(R).

Proof. In the proof of Theorem 6.9, we saw that tnp, = S"i, i is real-analyticon ff , and B-t(p) -

lwp,), which shows the first assertion.Let us prove the second statement. Denote by D the set of all points [S, /] g

?(R) such that [^9, /] is represented by a real-analytic quasiconformal mappingof .R to ^9. Let [So, f,] be an arbitrary point in D. From the first assertion,we find a neighborhood LI of the base point in "(Sr) so that every point of[/ is given by [S, /] with some real-analytic quasiconformal mapping /. Then1[S,f"f,] | [S,/] € U] is a neighborhood of lSo,f, l in "(n) and contained inD. Thus D is an open subset of "(,R).

Next, let { [S",.f"] ]Lr b" a sequence in D which converges to a point [S, /] e"(,R). We may assume that each /, is real-analytic. Since [^9,, f^of-tJ converges

taEaar 'seues elqnop uo rueJoeq? (ss?rlsrale1t uror;'*g uo crqdrotuoloq sr tdor. acurg

(o'g)

(e'g)

araq.n 'g (- ?' se c Jo slesqns lceduroc uo dluro;tun

(7)o a (z)lnlp?* z - (r)'n*

eler1 a^\ '19'y ura.roaqa 'tg'loo.t4

*H ) z,t fip%?.[ ;- =e)ta]06

fiq

uaal6 sr pu, ,7.rrr, lrtfo1aatTonu?p eql'(.t'tt)g ) /7 fr,r?a? rol 'Ol'g uraroaq&

'0 = tt ereqr$ es?t aql JaPlsuoc airn '1srtg'II'9 ulaloaq; ur ue,rt3 sr

[n]nO lo uorleluasarder ler3alur pu" ef,ua]slxa aq;, 'uol]?as Eurpacerd eql uI turou-oo? crloqredfq eq1 o1 lcadsar Wlu ecuaS.raluoc rurou st acuaS.re.tuoc eql areq^r

'((,t)o- P|d: ?-fi' =[nlnet

,tq paugep q [n]dO ueqJ '0 {- I se g..- -ll(l):ll P*

'("1'g)g

o1 sSuolaq n araq^r '(l)al + r1t + d' - trl leql qcns ur8rro eql Jo pooqroqq3rau

e ul I reqtunu xalduroc fue ro; paugeP t(t'n)S uI lueruala ue aq trl 1a1'Q'n)g :l z pue r(J'n)g I r/ fre.rlrq.re

ro; r/ 1e z uorl?arrp eqt ul O p [n]dO e^IlsAIreP eq] eugoP of qsIA{ arrr 'lro11

uollcaford 6srag Jo uor+Bl+uara:gT CI 'z'7,'g

'6VT,-ZV1, 'dd'[Og-V] 3eplo {ooq eq} ol reJar ear's1te1ep.rog'uap,tcrg o1

enp pu€ elr€g ol enp ere qf,rqrh ueJoeql qqt o1 saqceordde reqlo ere eraql'([6] srolgy prr€ 'gtI-IgI 'dd'[e-V] sroJIr{Y aas) (;)a;

ur lurod fue ;o pooq.roqqSrau ? uo O Jo uoll?as pcol crqdrouroloq " ltnJlsuoco1 elqrssod sr 1t

(uorlceger leruroJuocrs"nb Sursn ,tg 'uorpes

11a1y-s.tol1qy eq1

pallsc fl qc1q,r,r '(;)a; ur lurod as€q aqt Jo Poollroqq8rau e uo 6rt <-+ a1 uorlcas

pcol crqd.rouroloq e seq O uorlealord (srag 1€ql selsls 6'9 rueroeqJ, 'E ,lroue[

tr '(U)Z of lenba aq lsmu O'palaauuoc q (U),2

ecurg'(g)gJo tasqns pesolc e q O ecuaq Pue'6'ol s3uolaq [/'S] 'snql'(U)J

ur [u/orJf 'S] = [/',S] Pot cl$pue-par sr "/o,lf snqtr'u e3.re1f11uategns due

.ro3 ["f',u5] = fr_to"t,".g] q]l^ "s o1 s;o ud Eurddeu l"uroJuoarsenb c1ld1eue

-leer s slswe eJeql 1eq1 saqdurr uollrasse lsrg eql '(,S); lo lutod eseq eql ol

uoy?-ffo,^"[[ !r- = e)v]q

L9lacedg .ralpurqrleJ Jo ernlf,uls xalduro3 Jo ef,u"rn uI 'Z'9

158 6. Complex Analytic Theory of Teichmiiller Spaces

* ' r r = I * t i l t f u ] ' + o ( r ) ,- '1 , ,= t rbf r ] " +o( t ) ,t '1 , ' ,= t rbfv)" '+o( t )

uniformly on compact subsets of .F/* as I ---+ 0. Thus we see that

iD(pr) = {up, , z } - t t t fv)" ' + o( t )

uniformly on compact subsets of f1* as t - 0. Since f/./f is compact, it followsthat iDs[v] exists and is equal to rblv]"' . Further, formula (6.9) provides (6.8). tr

Theorem 6. 'LL. For eaery p e B(H, f ) t and v e B(H, f ) , the der iaat iue Qr[u]erists and is giuen bg

ouv'11'v=f-* lLmaea,t]-u1"1'�, zeH* (6 10)

Proof. Set f = *r, !11 = wprowrl , and \t = pcr.Then we have

A , ( c ) = ( L f ' - , ) " r - , ( ( ) , c € f ( H ) .' - \ E r - F p 4 ) " '

Thus, putting

\ / l \ / f " ' ' \) (o= \ ;Ld"r- ' {c) 'we get

)t = l ) +t6(t) on /( f1),

where l16( l) l l - - 0 as I + 0.On the other hand, from the relation

O ( p , ) ( z ) = { s * f , r } = { g t , f ( z ) } . f ' ( r ) ' + A Q t ) ( z ) ,

we obtain

ib,1,11,7= hi"r 1{n,,tel}1 f,k)r, z € H*.L r + u , I

Then by the same argument as in the proof of Theorem 6.10, we see that @r[u]exists and is represented in the form

f tibu1,11,1= l-9 [ [ -fg-'.,n alartl y'e')2, z e H*., " l l 1q1 (C- fQ) ) ' ' ' J " ' '

Therefore, by substituti"g /(O for ( in this integral, we obtain the integralformula (6.10). tr

sr C acurs '(.t)al> ('*DA lurod fue Jo pooqroqq3reu e ur crqdrotuoloqlq sl d1eq1 arro.rd o? peeu am'crqdrouroloqlq 8I r-go*(rr))org

- d }€ql aes oa'too.r'4

. (r,t)al, - (J)sJ : r_Ao* (o)or1,(Il)dt * (1)0a :*(o)

,(tt)t* (1)a:.la'l,(tU)Z * (U)Z ,.[V]

:ctrld.totuolorlrq nD eto sfutddput. 6urmo11ol eqJ 'ZT'g uaroaqJ

.(J,H)zV p (J)sJ ureurop papunoq € sepazrleer q (r"f)dZ ueql'(IJ'H)"V o1u1 (y)dglo Surppaqura (sreg eq Ig ta1

'7utod asoq ay1 to uotyolsuDrl e *(o) .ro *[o] lec osle elA

'ln*7 = ([nt]).(t),--, [n ̂]

rq (I.r)d,Z + (1)da:.@lusrqdrouroa{uoq e euuep elr'ralroaro141 'r_mod(nora - nn Ir-qI qtns pu€'oo pue'I '0

Jo qc€a sexu r_oott(nolo l€q} Ilsns uollsruroJsuerl snlqotr{ IeaI € sI D arel{l!\'l.n^l = ([n.])'[r] - lanl

:(lt)t - (.7)g :.[o]ursrqdrouoauoq s secnpur (tU),-U -- (U).f :*ltl) Turod ?saq eql to uotyo1suo"r7eq1 'flerrrloadsar '(t

7)"6'(,1),2 qtt^ (tU)-f '(g)g ,(gluapr e^\ uer{11y r-nJn - rJ

1€r[] eunss€,teur e11 'IU *-g:tf Surdderu leruroJuocrsenb eJo l;ll € eq t't taT'{pp} - r.7 ur luaurela alqe}rnse fq pexg sr oo pue'1 'g slurodJo qcee t€q} qcns3 snue3 Jo rU. a?€Jrns uueruar11 pesolc reqloue;o t7 Iepour u€rsqrnd e e{€I

.J JO

acroqf, eqlJo luepuadapur sl'(J)sZ * Q)dl,:g Surppaqure (sreg Sursn,tq g'1$ur peusep se^r qf,rq^\ '(;);

lo ernlcnrls xaldruoc eql l€tl1 a,rord 11eqs a,t. '.nog'1 reldeq3 'ltt-Vl srrcpox pue rrorrol{ pue 19

raldeq3 '[Og-V] srrr€H pue sq]lgrJ9 o] reJer aa,r 's1e1ep rod 'uo os pu€ 'sp1o;rueur

xalduroo uee.&\1eq s3urddeur crqd.rouroloqlq pue ctqdrouroloq'p1o;tueur xelduoc" uo suorlcun; crqdrouroloq'sp1o;rueur xalduoc leuorsueurlp-u eugep uec arrt'1raldeq3 Jo I'I$ ur sploJrusu l"uorsuaurp-euo Jo es"f, eql ur sy '(6'd 'hf -V]

srag aas) dlrnurluoc s1r sarldurr ,f go ,tlrcrfleue aleredas leql slresse uaroaql

,sEo1re11 (raqlrn{ 'f1a1e.redes alq"rr"^ qf,ea ut crqdrouroloq pu€ O uI snonul}uo)

sl / 1€ql pepr,ro.rd 6. ur erqd.rouoloq sl / lerl? uaas fltsea sr 1t 'elnurro;

1e.l3e1urs,dqcnep dg 'o yo pooqroqqErau s ur (ur'"''rz) = z IIe JoJ saS.raruoc qctq,u.

.r(uo - "r)' ,r(ro - rz)"q 'tq:,t{''

T= Q)l

uorsuedxa sarras rau,od s s"q tl'O ) (up' "''tD) - n .r(.ra,ra roJ JI O uoctyrl.totuoloy pell€? sr uC Io CI ur"ruop e uo peugep 3l uotlcun; panlerr-xeldtuocy 'uorsuermp raq3rq Jo splo;tueur xaldruoc fgarlq /rarleJ ero, 'q1t.tr ut3aq o5

(,f)^Z f" arnlcnrls xelduro3 Jo acuerrBlul 't'Z'g

acedg ralnurqrlal Jo ernl)nrls xalduro3 Jo af,u"rr"^ul 'Z'g

'r{

69I

160 6. Complex Analytic Theory of Teichmiller Spaces

homeomorphic. Let f p = wpl(dr)-l and Bu be Bers'embedding of TB(f ts).If.

two mappings

f i -Bo ( (wP) - t ) . oB ; t : TB( IP ) -Ts ( f ) ,

Fz =B; .o(uo( . t ' ) - r )*oBpL : Ts( f P) * "s( |1)

are biholomorphic in a neighborhood of the base point of Ta(fP), then .F =

FloFlL is biholomorphic in a neighborhood of B([trlr]) eTBQ).Thus, it is sufficient to give a proof that F is biholomorphic in a neighborhood

of the base point of TaQ).Take a neighborhood, say V = {9 e A2(H* ,f) | l lel l- < 1/2}, of the base

point of Tn( l ) . For arb i t rary p, t €V,we set D = { t e C I tb+t9 € I / } . Putp(t) = t+t9 and p(t) = Its(t). Flom Theorem 6.9, we have B-'(p(t)) -

[urr1tyJ.Let )(t) be the Beltrami coefficient of wr(t)ou-t, which is given by

. \ ( r )_ ( r " r ( t ) -u , \ ^ , . ,_ ,^\ot - \q t -F-*{a )"-

on f/. By the construction, we get f(e(t)) = {ror(r),2}. Since,\(t) is holomor-phic with respect to t, Theorem 6.11 implies that F(g(t)) is holomorphic on

D. Since g and ry' a.re arbitrary, .F is holomorphic on V. Since .F' is injectiveon ?s(i-), from the following lemma (Lemma 6.13) and the inverse mappingtheorem, we see that .t' is biholomorphic on I/.

By the definitions, the rest of this theorem is trivial. O

Lemma 6.L3. The Jacobian "Ip = det(0Fi/0zp)4i,1'=n of an injecliae holo'morph i cmapp ing F = (F r , . . . , f ' " ) o f a doma in D i nC" i n toCn aan i shes a t

no points on D.

Proof.We prove this a.ssertion by induction for dimension n. First of all, clearlyit holds for n = 1.

Given an integer n ) '/.,,

we assume that the assertion holds for any positive

integer S n - 1. Let Do be the set of all points in D where .Ip vanishes. We wantto prove that Do is empty.

Suppose that Do is non-empty. F\rrther, assume that the Jacobi matrix ofF is of rank r with 1 S r S n- 1 at some point a € Do. Then we maya^ssume that det(d.Q /|rx)1.5i,x9, does not vanish at c. The inverse mappingtheorem impl ies that G(z) = (F1(z) , . . . ,Fr(z) ,zr*r t . . . ,zn) has the inversemapping H = (Ht,. . . , Hn) in a neighborhood of a. Then we have F"+r(() =

G + t , . . . , F . F / " ( O - ( ' , m d F o H G ) = ( ( r , . ' . , G , F " + r o I / ( ( ) , . . . , F n o H ( ( ) ) i n aneighborhood of G(c). We set

W = { ( = ( ( r , . . . , ( , ) € C ' | ( r = F r ( o ) , . . . , G = . F " ( o ) } ,

Wt = {w = (w r , . . . ,wn ) € C ' I w r = F r (a ) , . . ' , u r = t r ' ' ( o ) } .

Then the restriction FoHllat is an injective holomorphic mapping of a neigh-borhood of G(a) in I,7 into Wr.By the hypothesis of induction, it follows that

'8'I $ ul

pa?npur auo aql o1 luap,rrnbe s1 (.i,)dg uo ernlcnrls xalduroc stql 'ecue11 'Q)aJ,

ol (J)dJ;o Surdderu crqd.rouroloqlq e q g Suppaqrue (srag 1eql reelt q ?I'(t)d,1, uo arnlcnrls ploJlrreu xelduroc e se,rt3 {(.t)d,t > ln^) | ("nlnl'ntl)}'erogaraq; 'p

+ nn U nn q1r,u rp pue n2 f.rcra ro; rtqdrouroloqlq sI

(nnUnn)nt *_ (nnudn)df : r_(d,{)on,{

l€q} s^irolloJ 1l 'zI'9 tuaroeql ,tg '[nr] Punore pooq.roqqtrau e]eulProoc

e se ("n1,tg 'nn) n{rt uec et$'acua11 'd4 oluo /4go usrqdroruoeruoq e secnpurnd por '(,1)d.t ul [/.]Jo pooqroqq3rau e s1 d4 pqt s?ress" g'g uraroaql ueqJ

'(ntn)r-(nd) ='n pu" {zlt > -lldll | (rJ',H)zv ) d} = nA

+esal111\ '(rJ'*H)zV o1q (n.i,)dggo Surppaqura (sreg sr dg araqn 'g = ([dnt])dg qlrm

G,t)st * (.t)dt : *(nmlod g - ag

Eurddeure,rrlcafur ue a^€q aar 'r-(no-l).ir {n = tJ 3ur11as :snolloJ n (l)g,t > ln*] lutod fue

Jo pooqroqq3rau aleurp.rooc e e)p+ ea.l, 'f1aure11 'lelluesseur ,tlen1ce fl ploJllrelupcrSolodol leuorsueurp-(g - 0g) I"er € sr (l)gl, leql lc€J aq1 'ctqd.rouroloqlq sIg leql qcns ernlonrls plo;tueru xalduroc e seq (.7)dg l€ql ees o1 'ralarrroll

'crqdrouroauroq sl (J)sJ -- (l)dl,:g Surppequre (sreg l€q1 aas ol sureuopJoacu€rJe^ur uo rualoeql s(JeA{norg Pasn a/$ acurs'p1o;rueur parEolodol leuolsueurP-(g - 0g) I"er e sr (l)dt leql /$oul ol pepaeu aarr 'eraq; '(J'

*H)"V aceds rolcaaxalduroc eql ur ur€ruop pepunoq " $ qcrq^{ '(t)s,t uo arn}rnrls xalduroc aq1r.uorJ pernpul seo, (;)dg uo ern?f,nrls xalduroc eql'g'I$ ur pelsls sy'tlrautey

tr 'u ror sProq s."."er eqr acuaq oT"i,il}i,T ffifi;"_;:':iil:*",,

qcrqar'1ue1suo? " aq lsntu .!,{rarra'acua11 'In uo saqsrrrel ttg/lge fra,ra'06,

uo qsrusA tzg/lgg 1e acurf 'Ln uo uorlcunJ crqd.rouroloq e s€ pereprsuo) sIq)lq^{ 'n

U oO ol .{4';o uorlcr.r?sar aql aq fd fe1 'rC u-r ul€ruop elq€?rns € sI

n araq^\ '{tn > " | ((t)f'r)} = nUoO 16q1 qr.,r 1-,C 4 Ip ureuop e uo

4l uorlcun; crqd.rouroloq e slsrxe araql leql arunsse deur er,.r'(07'd'[ft-y] srag)ueroaql uorlerederd (sserlsJaralt urory 'g uo if uorlcunJ crqdrouroloq eql Josoraz Jo 1as a{l sr 06r acurg 'O

Jo lesqns radord e 4 oO 1eq1 aurnsse '1xe11'O uo e^rlcelq q A 1eql slcrp"rtuoc slqtr'O uo tl$uel^ tzg/lgg

IIe esneceq '1ue1suoc e aq lsnru fg qcee ueqJ 'O = oO l€q? arunssr 'lrolq'oO uo qsruel plnoqs tzg/lgg i(ra,ra snqa'I -u t {u€rJo sl J leq} slrrpsrluocqcrqin 'o

le u {u€I Jo aq lsnur dr Jo xrr}€ur rqo?ef aql 'p Jo poor{Ioqq3rau e

ur crqdrouoloqlq q 5t ecurs 'pu"q reqlo aql uO '(r)g tr u {uer Jo s! IIoJr Joxrrl"ru rqocsf aq1 'acua11 '@)C t" r - u {uer yo sr zltlgog Jo xrr}etu rqocef aq}

I9Iacudg ralpurqrlel Jo ernlf,uls xalduro3 Jo a)u"u"^ul 'Z'g

162 6. Complex Analytic Theory of Teichmiller Spaces

6.3. Teichmiiller Modular Groups

We shall prove that the Teichmiiller modular group Mod(,R) of a closed Riemannsurface of genus C(]=2) acts properly discontinuously on the Teichmiiller space"?(r?) as a subgroup of the biholomorphic automorphism group Aut(f@\ ofr@).

6.3.1. Definition of Teichmiiller Modular Groups

Let.E be a closed Riemann surface of genus C(22). We define the Teichm[illerrnodular group Mod(R) ofR as the factor group ofthe group ofall quasiconformalself-mappings of ,R over the normal subgroup of those homotopic to the identity(cf. $3 of Chapter 1). The element oI Mod(R) defined by a quasiconformal self-mapping f " of R is denoted UV [/,].The action [/,], of an element lf "l e Uoa@)on ?(r?) is given by

[/,].([S,/]) = [S, f"f" ' lfor every [S,/] e T(.R) (see $1.3 of Chapter 5). We call such an [/o]. aTeichmilller moilular transformation oI T(R).

Let f be a Fuchsian model of ft. By lifting, a quasiconformal selfmapping "f,of rR corresponds to a quasiconformal self-mapping ar of the upper half-plane I/with c..,fc.r-l = f . Let or; be a lift of a quasiconformal self-mapping f; of r? withu;f (u;)-r = l- for i = I,2. By the same axgument as in the proof of Lemma 5.1,we see that [fi] = [/2] in Mod(R) if and only if u2 = ca1o7, holds on the real axisR for some jo € |. With this in mind, two quasiconformal self-mappings {rr1 andu2 of H satisfying u;fulr = f U = 1,2) are said tobe equiaalenl if there existsan element 7o ol I such that e2 = uroTo on R. Denote by [c.r] the equivalenceclass of ar. The Teichm'iiller modular group M odQ) of f is the group of all theseequivalence classes [a.']. The action [c.r]- of an element [w]e Mod(f) on T(l-) isgiven by

[r]- ([ru]) = laowq ow- tf

for every [ru] e 7(f), where a is an element in Aut(H) such that eorlrou-rfixes each of 0, 1, and oo. (cf. $2.3). Theorem 6.12 asserts that [cr]- is a biholo-morphic automorphism of ?(f ). Furthermore, this [c.r]* induces a biholomorphicautomorphism (c,r),of TaQ) defined by

(r).([ru]) = lw,)

for any lrul e TB(f), where z is the Beltrami coefficient of oowrou-l . We usethe same notation (cl). for the biholomorphic automorphism ol Tp(f) insteadof Bo(w),oB-1, where B is Bers'embedding of Tp(l). We also call [ar]- or (c.,)*a Teichmiiller modular transformation. By the construction, it is obvious thatModQ) is isomorphic to Mod(R).

By the identification of ?(f) and "(rR) (Proposition 5.3), the Teichmiillerdistance on 7(-R) induces the Teichmiiller distance on ?(f). Then Proposition5.5 implies the following.

',t1duns aroupagrre^ q tnq

'ZI'g rueroaql u€ql ra{pellr sr q)rq^r 'l1nsa.r qcns raqlou" ^roqs lleqsa,r'are11 '(61'gure.roeql'Jc)

UJo J Iepour u€rsqcnd pezrl"urrou€ sa)npulU uoscrsepoe3 pesolc Jo sqfuel crloq.redz(q;o les e '61'g ureroaqtr uI u^roqs s€^r sV

'L to aco.t7 aqy to a.tonbs

e?il sel.ouep (L)"r1e;eqn'1gur a1atcslp s! {J ) Ll (t)zrt} ??s eqJ'fre11oro3

',t.re1oroc 3urmo11o; eql ol peal g't pu€ gI'g suorlrsodor4

o 'gl'g eururarl slcrperluoc slq,L'u .{ue ro3 W > (il1 = (Q)uL'z)d 8)'urut

leql qrns JJo sluauele lcurls-rp,tlen1nur;o ecuenbas e q r?{ u,0} uaql'u,[3oslx"e eql sl "ty eraq,u'p *"tV Ud ttsql Pue'(8 reldeq3Jo 8'I$ eas) "7 srelos",1 leql os .1, ) u,L luetuale u€ e{sJ 'g ut leeduroc ,(1errt1e1a.r sI qclqrr J roJ .4ur€urop Ftueu"punJ e esooqC '"I

Io t{fua1 cqoqradfq eq} sl ("7)/ araqrrr ',tr4l

raqurnu aarlrsod aruos roJ W j (l)t seg$les u7 ,trerle leql qcns g' uo scrsepoe3pasolc lf,urlsrp fgenlnur Jo r=.:{ u7

} ecuanbes e slsrxa eraq} ?eq} aurnssy /oo.l2,

'7 y76ua7 ctloqtadfiy IlWn A uosctsapoa0 paso1c fruoru fipyru{ Four ID Isrre N,eq?'7 aa4tsod fiuo .tol 'taaoano147

'E u, ?pns?p s! Q,7) 6 snua6 to g acottns uuDureNy pesol?D uo s?r,sepoaf pasolt 1r'o to st176ua1 eqoq.radfiq to 1as eqJ 'gT'g uorlrsodor4

tr 'uorlslPeJl

-uoc e'(n)lnv;o dno.rEqns elerf,srp € lou $ J'gI'7, euurerl ,(q snqJ,'Q1)nVo1 s3uolaq / feqt srrrorls 8I'Z €rrrueT'17 ul sl orn acurg'oot = (oz){ ateqell^uer{J 'Il uo fprue; I€Lurou e fl lI asn€caq '.,ir1, uo peugep / uotlcuny ctqdrouroloqe ol sles lcedruoc uo fpulogrun sa3ra,ruoc I*{ "f

} teqt etunss€ osle .{eur arrr'.raq1rr1g 'oo + u w H ) om <- ("r)"L prc )1 ) oz +- "z leq+ eurnsse fetue,rl 'r(resseceu;r acuenbesqns e Eut1e1 'alalduroc sr d pue lceduroc $ I4 etuls'X ) "z auos roJ W j (("r)"L'"2)d sagsrles u,L.r(ra,r.e leql qcns J uI slualu-ele l)urlsrp ,t1en1nur Jo I1{ u,L

} acuenbas 3 slsrxa araqt }€tl} asoddng /oo"l4'

'H uo e?uolsrp ?rmurod aqy st d a.taqm 'ytr

j ((r)t'z)d Y)'uutt q??n J ) L fiuout fr.1a7tu{ lsoluu ID lswe ere1l '14J taqu,nu aatT-tsod puo 11 auoyd-l1ot1 .r,addn ayl u, >J psqns Tcodu.toc fitaaa "tog 'gl'g Btrrwarl

'suorlrsodord

auros e.reda.rd em '(.7)"6 uo rllsnonutluo?slp fgedord s1re- (,1)poq 1eq1 a,rord o;

slas qnpotr I 'z'8'9

'ecuDlsNp rellnuqral eyq oy Tcailsa"t qTtm fi.tyau,os, uD sN (l),f, lo *lnl tustrld

-routolnD ctyd.r,otuo1otllg eW '(J)poW > l'r,l lueuep fi"raaa tog '?I'g uraroaq5

t9lsdnorg r"lnpol4l ralllurqf,raJ't'9

164 6. Complex Analytic Theory of Teichmiiller Spaces

Proposition 6.17. Let f be a Fuchsian model of a closed Riemann surface ofgenus g (22) . Let { t i }T=, be a system of generators for I suchthatT has the

repelling fired point 0 and the attractiae fixed point oo, and such that 72 has the

repelling fired point r with r < 0 and the attractiue f,red point I. Then each 7iis iletermined bE lhe absolute ualues of traces of elements in the fi'nite sel

g = { 1 p'l z, 7 j, l ft ol x, ̂ tt ' oj r, ( lro7r)*' o1,r },

w h e r c j - 1 , . . . , m , a n d l c = 3 , . . . , f f i .

Prool. First of all, we may a^ssume that 71 has a matrix representation

, f ) o I' = L o ' l l ' l ' A > 1 '

Thus 71 is determined by the absolute value of tr(lr).Next, by the normalization condition we may assume that y has a matrix

representation

Thus we geta * d, = l t .(rz)1, a.\ * d)-r - l tr(71"72)1.

Hence both o and d are determined by the absolute values of traces of 7r,72,and 7p72. Since the quadratic equation l2(z) = z has a solution 1, we have

2 c = a - a + 1 f t 1 a 1 2 - 4 , b = c * d - a .

Consequently, both D and c are determined by c and d. Therefore, 72 is deter-

mined by the absolute values of traces of 7r,'Yz, and 71o72.Now, for every [ - 3,...,rn, the Mobius transformation 7; has a matrix

representation

" = l l : ] , P , e , r , E € R , p s - q r = r '

Here, by Theorem 2.22 we may assume that

P + s > 2 '

Then, using the relation

tr(.a)tr(C) = tr(AC) * tr(.4- rC), (6 .11)

a = l o ' , . | , a , b , c , d . ) 0 , a d - b c = 1 , a * b = c { d .L C d J '

we see that tr(,AC) is determined by the absolute values of traces of lt, 7*and 7fo73. In fact, since the left hand side of (6'11) is positive, we have

tr@Cj = ltr(,aQl - ltr(71o7r)l provided that ltr(,Ac)l ] lt.(e-'q)|. tt

It;(/i)l < ;tt(e-rc)1, then *" oLtuin ft(A-tC) = 1tr(A-rc)l = ltr(7!r"7i)1,and hence tr(AC) = ltr(7r)l . ltt(z*)l- ltr(7r-1o7j)1.

ulstqo ar$ snql ',L o1 seEra,ruor I?{ 1-(rloo"o)oLo(rlnouo) } acuanbas

aql 'J =l Lt.ra,re roJ ?"ql aes a^{ '(,t)z ol [pl] ot se3re,ruoc I;i{ [rloo"o1 1eturs 'oo pu" 'I 'g go qcea saxg ,loouo leqt qcns (g)lny 3 uo ereq^{

'[1loo"n1 = ([pp]).["]

;o uo Eu t dd€ru -JI es Ie urloJuo cI se "o "-:::;t,"f t ":i j"; ;; \7"'lft: l:T

"'[prJ ot sa3ra,ruoc r-*{Wl]"rl

} feUterunssr feu aar 'od olur (,f )Zfo lurod aseq aq1 Eurlelsuerl 'acue11 'od o1 sa3ral-uoc Il{ ('d)"rt} ttql 'od - (ob)o6 = ('d)'rt 1aE aal '("d)"Sor+"6 - ("d)"Ufg'od - (n)'0 ulelqo en'ud - ("a)uSou6 uro14 'rtlaarleadsar 'oq pr* od s$urd-deu crqdrouroloq ol (J)J ur slas lceduroc uo fpr.ro;run aS.rarruoc r*{ 'rl

} prtt}{"}',t1rc1yu19 'ob = (od)ol eleq e,a{'relncrlred u1 'of Eurddeur crqdrour-oloq e ot (J)J ur slas lceduroa uo ,(puoyun sa3raluoa r-/{"t

} te{t erunsssfeu aaa 'fressacau y acuanbasqns e 3uu1e1 'snq; 'f1rure;

l€rurou B sl r?{ T }

'(g'g uraroaq,f,) (.f)s,J ursruop papunoq e o1 crqd.rouoloqrq sr (;)g ecurg

'utorlufi - uq 'rlI = "6

les e^\'u qcse IoJ'Q),f,> oD lurod e o1 sa3ra,ruoc r?{ ("d)"{ } acuanbas eql '(J)J

3 od lurod e o1 sa3raluoe q?rq/r ("f)'f q ,-*{"a } acuanbes ur€lree e ro3 'leqt

qans (.2.)po7g ur sluetuale 1?urlsrp flenlnur Jo I=g{ $ } acuanbas s s}srxe ereqlueql '(J)J uo flsnonurluo?$p fl.redord lce lou seop (1)po1,tg 1eq1 asoddng

'(6 reldeq3 Jo g$ ul {r€urag aas) 11'g uorlrsodor6ur srsaqlodfq aq1 Surf;slles srolereua3 ;o urelsrts e seq J leql IFcaU

'too.t4

'(Q)Dt"vdno.tf rustrlil"rotuolno nyd.toutolotl?q ?W Io dno.tfiqns o so (1)a uo filsnonutTuoc-stp fiy"tado.rd sTco (.1)po141 ilno"rf l,o1npou reIInurUoNeJ eUJ '8I'g uraroaql

sdno.rg Jslnpor4l rallnruqcral, yo z(lrnurluocsrq' g' g'g

tr 'qLo+(zLorL) pue'lLofL'ttolL (qL (eLorL(zL 'rL

Jo sarcrl Jo 6anl"A etnlosq" aql ,tq paunuretap st {,L }e{l epnlcuoc e^r

'r_ysp * r_yDa + yrg + ydo = (CaV)rt'sp+bc+rq+do-(gg)r1

'r_ys*yd=(gy)r1's 1d - (g)r1

suorlenba reaurl Jo uralsfs eq1 3ut,r1og''tLo+(?'LorL) pue '{1, (zLorL

Jo saf,erl Jo 6enlp^ e}nlosqe eql ,tq peuruJelepsl (CSy).rt 'f1.regu4g 'cto*L pus '{1, 'zf

Jo sare.rl Jo senle^ elnlosq" eqt fqpaurrurelep sl (CA).rt 1eql aas am'y;o pealsur g Sur.reprsuoc'feaa eures aq1 u1

99Isdnorg r"InpoIAI reillurqrlal 't'9

166 6. Complex Analytic Theory of Teichmiiller Spaces

since { t,2(z) r,, . ;;;,i;":l;" "i,];J,'.;..,*ition 6 16), andsince every wno7ou|r belongs to f, we have

tr2(uf , ro1ou,) = t r2(7) , I eg

for every sufficiently large n, where g is the finite subset of l- given in Proposition6.17. Hence, Proposition 6.17 implies that for every sufficiently large n, thereexists an element 0" e AutlH ) such that

unro1oun - | ito',to\n, I e l.

This shows that Bn belongs to the normalizer N(f) of f in Aut(H), and [c.r,,]* -

[f"]-. tfrus every such [c.,,]* fixes the base point [fd] of T(l-).By the definition, it is easy to see that the isotropy subgroup of Mod(f) at

[id] is isomorphic to N([)lf . On the other hand, it is well known that N(l-)/fis isomorphic to the biholomorphic automorphism group Aut@ lf) of the closedRiemann surface H f f , and that Aut(H/f) is a finite group (see the followingRemark 1). Therefore, { [ar,,]* ][1 should be a finite set. This contradicts that

{ f" }T=t consists of infinite elements. D

Remark -1. Every element o € N(f) induces a biholomorphic automorphism [o]ot Hll defined by [a]([z]) = [o(z)] for any [z]e H/f .It is easy to see that themapping a r* [a] is a homomorphism of N(l-) onto Aut(H/f) whose kernel isl'. Thus N (f) I f is isomorphi c to Aut(H / f).

H. A. Schwarz proved that Aut(H/i') is a finite group. F\tther, A. Hurwitzshowed that the number of Aut(Hlf) is not greater than 84(9 - 1). For thesefacts, we refer to Farkas and Kra [A-28], p.242; Siegel [A-98], Vol.2, p. 91; Tsuji

[A-108], p. 496; and Imayoshi [104].

Remark 2. In the proof of Theorem 6.18, we have used the fact that "(.1-)is biholomorphic to a bounded domain. However, we can also verify Theorem6.18 from Theorem 6.14, i.e., the fact that [r.r]. is an isometry with respect tothe Teichmiiller distance on "(l-) for all [w] e Mod(l-) (see Gardiner [A-34],$8.5). Moreover, by using a theoremof Nielsen on topology of surfaces, we canshow directly that M od,(R) induces a discrete subgroup of the biholomorphicautomorphism group of 7(,t). The proof of this kind is in Nag [A-80], $7.1 ofChapter 2.

Now, we have the following fundamental theorem on the moduli space Mn.

Theorem 6.L9. The moduli space Mo of closed Riemann surfaces of genus g(2 2) has a norrnal compler analytic space strtclure of dimension 3S - 3.

This theorem is an immediate consequence of a theorem due to H. Cartan

[48]. Namely, for a given discrete subgroup G of the biholomorphic automor-phism group of a bounded domain D in Cn, the quotient space DIG has a

W ) b 'd ' (b 'd) n{,p } (b 'a) ,l{,p

pue d - od r{ly* w > ,d'"''od,slurod IIe ra^o *orrt;tffifdtJ,i;';#I=!

r 1t4, r1d)\pT r", = (b, Q n{p

1nd arrr 'u .ra3alurarlrlrsod ,tue rog 'b = (ilt pue d - (r)/ qtp W - V :rf Surddeur crqdrourolotle slsrxa eJaq? t"ql qcns y ) g'o slurod IF ra^o ue{el fl rumuuul eq? alaq/{

'(q'o)d lur

- (b'd)\p

les e^r 'W ) b 'd slurod ollr1 ue^l9 'ploJlueur xalduroc e eq W pj'ploJrueu xalduroc

preua3 e oI V ,srp lrun eql uo d acuelsrp eJe?urod eql Jo uorlezrleraua3 e srqcrqan 'acuelsp rqser(eqox aql ecnporlur a/rr 'sureJoeq} s,uap,tog eqrmsep oJ

'uapfog ol anp osl€ eJ€ qcrq,rl (66'9 pue

IZ'g $ualoeq;) suraroaql Eur,rlo11o; eql f,q pe,rord sl *? Jo ,tlurlcafrns eqa

'(tZt-gZt'dd '[OS-V] 8ep aes) Z - 6 +r'q1 peprno.rd e7 ol cr.rqdtoruosr sr

+r;o 1aura1 aql l€ql s^r\^.olloJ ll 'lce; srql uord 'C ralo eceJrns 3ur.re.,roc paqcu€rq

palaaqs-orlrl e ,{q paluaserda.r sr U 'sl

leqt 'cr1-dr11a.raddq sl ZI acurs ,orll1 .rap.ro

3o rusrqdrouolne rrqd.rouroloqlq e seq oar? snua3 ;o Ur er€Jrns uueruarg pasols,{ue'pueq raq}o eq? uO'e < t }eql papuo.rd earlcalur sr *z ecuaq puts ,{

[pp] ] of

lenba sr *? Jo leura{ aq} snql '(1 uorlrsodor6 o1 frelo.roo ,[6lI] q?n€U pue ,[0gI]

sapr/{arl :gZ,Z'd '[Og-V] srrreg pue sqtgrr5 aas) {pz } = (A)?nV qll,!\ gr ereJrnsuueruerg pesop e slstxe eraql l€rll u^{oDI sl ll'e 4 f .rog '(,DpoW ) [o] ,(.re,reroJ +[rn] = ([r]).1 fq ((.7)g)nv ot (J)poW;o +r tusrqd.roruotuoq € auuep eM

(uapfo11)'0e'9 uraroaql

'[gzt] ntx pue '[gg] €rx pue a1.reg '[?t-V] .raurpreg'pg1] uep,(og aas'sgelap rod 'llnsar srq ureldxa fger.rq II€qs eM 'drqsuorlela.r Surr'ro11o; eql e^eq(;)g eceds rallnurqcrel er{} Jo ((l),Dl"V dnor3 ursrqdrouro}n€ crqd.rotuoloqrqaqt pu€ (,ilpoW dnorE relnpou rallntuqclel eql leql parord uapfog .1 .11

stuoroaql s(uap^oll ?'g

'7 tueroeql'[6lt] pw '[911] qcneg eag'slurod .reln3urs setl d11en1ce (Z 7 0) 674J acedsrlnpou drarra 1eq1 uldou{ $ lI

'g raldeq3

Jo U $ '[gI-V] dgng ol reJar am 'tueroaql s(uelJeC ;o;oord e rog .crqd.rouoloq sr

CIO - O :1, uorlceto.rd eql teql qcns arnl)nrls aceds cr1{1eue xalduror leurrou

'k < o) (t)pow l�.ft= o) ez/Q)pow J

=\\J)J)?nv

L9lsruaroaqJ s,uapfog'p'g

168 6. Complex Analytic Theory of Teichmiller Spaces

for all positive integers n. The Kobagashi pseudo-distance d7,,1 on M is defined

bydu(p,i l =

"1!g {u@,q).

It is an ea.sy matter to show lhat dva: M x M -* R is continuous and sat-

isfies the axioms for pseudo'distance: dru(p,C) > 0, ilu(p,d - d1,a(q,p), and

dM(p,C) * du(q,r) 2 d,a(p,r) for all P,Q,r € M. It is said that d74 is non'

degenerate if d|,a(p,C) = 0 is equivalent to p - g. Note that du is not always

nondegenerate. For example, if M = C, then obviously du = o.If dru is nonde-

generate, dy is called the Kobayashi distance on M , and M is called a hyperbolic

compler manifolil. A hyperbolic complex manifold M is said to be complete if. it

is complete with respect to d1'a.

The most important property of dru is the distance decreasing property,the

proof of which is trivial by the definition: let M and N be two complex manifolds

and let f : M - N be a holomorphic mapping. Then it follows that

du(p,c) Z dv(f(p), fkD, p,q e M.

In particular, every biholomorphic mapping of a hyperbolic complex manifold

M is an isometry with respect to dya '

Theorem 6.2L. (Royden) Let rQ) be the Teichmaller space of a Fuchsian

rnodel I of a closed Riemann surface of genus c (22).Then the Teichmiil ler

dislance iI on T(f) is equal to the Kobayashi dislance dr1).

Proof (an outline). We give a sketch of its proof'

In order to prove dre) = d, it is sufficient to show that df,.y = d. In fact,

if dl(r) = d, then d|1"; satisfies the triangle inequality' Thus, by the definition

we have 4<n= d|,r, for any positive integer n, and hence d71;; = d.

For any [ru] € ?(J'), we put l-p = wqf(wq)-r, and denote by d, the

Teichmiiller distance on ?(i-p). Recall that every lrrl e ?(f) induces a biholo'

morphic mapping fwt)*: 7'Q) -fQr1(cf. $3.1).Now, to prove df,a, = d, it, suffices to prove the equality

di<r.>(i4,[r^]) : dt (lidl,lw^l), [r^] e T(f')

for any [trl,] e "(l-). In fact, if this equality holds, then the relations

it(lrul,[r']) = d r(1i4,[to'o(up )- 1]),

di < r flr r), [r' ] ) = 4 g, 1(li d), [u.'" o ( top )

- I ] )

imply that d(I.ul,[r']) = di<r>(1.'1,[to"]) for all [tue],lw') eTQ).

To simplify the notation,lp,dp, and u) in the above equality are rewitten

as f,d, and roP, respectively. Then, we shall show that

dl<n(i4,[.r]) S d(fidl,lwFl), lwq) e rQ),

.p =p teq? u^,\oqs sr tr uaql .(,f )Z ut

b pue d Eururol p sqled qloous esrmacard 1e ier.o ua)Fl sl runtugur eql ereq^{

,@)t fri = (b,d)p

1nd aal '(,t)l> D'd slurod oir,r1 fue .rog

'?p((t),c'G)c),t of = O)t

1J

las e^\'("1)Z *- [t'O] ,C qled qlootus asrarecard f.rerlrqre ue .rog (rr)'ooC sstspJo lou sI d teql uarou{ oEs sr tt ('[Og]

alr"f, osle aeg) 'uorlras oraz eql 3o slurod 1e ldacxa (.f); fo epunq luaEuel eqluo 1C sselt Jo sl ,t teql paUIreA sl lI 'I = Illolll q1,r ("J'H)zV 3 ol ge .rarroue{el sr urmuardns eql pu" 'H u\ nJ roJ ureuop leluaurcpunJ " sl n,{ erar{^{

,loo*o e)d> (z),-(^q"lrlnl - |

*1"" [ [ "rldns6 = (v,[,.]),4|

"'| V

"(nmt1 rr I

1eq1 pa,rord q lI 'p ?rJlalu Jallnuq?ratr aql uorJ pacnpur

(.f)Z f" elpunq 1ue3ue1 eql uo f,Irlatu leursalrugul aql se pareprsuoc sl 3' slql

3 o<r'o*r

G;WW urrl= (y'[,or])g

tas e,$'(,1'tt)g ) y pue (.f),f f [,ar] lurod due rog (r)'1 raldeq3

'[ru-V] reurpre9 pue 'F8I] uap{og aas 's1re1ep rog 'flqenbaul qq};o;oord aq?urc1dxa i(gerrq e11 'uapfog ol anp ;oold aqt go 1rcd l"rcn.rc lsoru eqt q sHI

' (0,r, > [an]' (lr*l'Wpl)p ? ([n']'W'4)Q)1p

a,rord o1 paau aa,r 'fleurg

'Q),t) la^l '(lr4'Wtl)p j ([n'] 'WtDQ)*p

a^eq era 'ecueH

ffi sor - (r'g)d ; (lr*J'Wp))op

= QD^!'W!l =(o),/ qq^ Burddeur crqd.rouroloqe sr [,nm] =*[j[it;]* iY;1-V:d/ Eurddeur eql l"ql aes e{'ldtl/$t - rrl Eurllas'pueq reqlo aql uO

ffi 3o1 = ([nrn] 'p.Pl)P

spled (6'9 ureroaq;) ueroeql ssauanbrun s.railnuqtrel uerlJ, 'I > { ; 0 '{

euos roJ Vl/{q - orl qt.r^ [,rm] = [",rrn] ]"qr qans (.7'H)zV I d luaurale ue

sl$xe aleql '(91'g ura.roaql) ruaroaql ecualsxe s(rallnuqcral ruoq 's^rolloJ se

69Isuaroaql s.ueP^o'u't'9

170 6. Complex Analytic Theory of Teichmiiller Spaces

(iii) It is verified that every holomorphic mapping f : A- ?(l-) satisfies

F(f (r), f '(,)) S :W r e a.

To prove this inequality is essential, though we shall omit the details.(iv) Take an arbitrary holomorphic mapping f : A - "(f) with f(") =

[fd] and f (O) - [rop] for some points a,b e A. Then (ii)_ and (iii) imply thatd(lidl, lwpfi = d(l id),[ru]) S p@,b). By the definit ion of d|r,rr, we get

d(lidl,fwp] 5 dlr.l([id], [ru]), fwp) eTQ).

This completes the proof of Theorem 6.21. tr

Now, Theorem 6.21 asserts that every element f e Aut(TQ)) is an isometrywith respect to the Teichmiiller distance on ?(f).

Take an element f e ,+".t1f1f)). For every p = l.ul € "(f), we set q =

f (p) = [tr']. The derivative f "f f at p is a complex linear isometry of Q(7(f ))to fo("(f)) with respect to the infinitesimal metric F, where Q(?(f)) and

Tq(I:(f)) denote the holomorphic tangent spaces of "(,l-) at p and q, respectively.

Here we use the fact that the dual space of fpgg)) is canonically isomorphic tothe space Az(H,l.p) of holomorphic automorphic forms on H lor lP , a fact which

is proven in the next chapter (Theorem 7.5 and Proposition.T.8). Similarly, the

dual space of foQQ\ is identif ied with ,42(11, f '). Hence, / induces a complex

linear isometry o of Az(H,1"t) to Az(H,f') with respect to the infinitesimal

cometric induced by the Teichmiiller distance d.

Here we know the following fact.

Theorem 6.22. (Royd,en) Let a be a complex l inear isomelrg of A2(H,lp) lo

Az(H, f ') with respecl to the infinitesimal comelric induced by lhe Teichm'ii l ler

distance d" Then lhere edsts a biholomorphic mapping htFll ' - HllP and

a compler number_c with lcl = 1 such that a(9) - caoh ' (h')2 for all 9 €Az(H,lP), where h is a l i,ft of h to H.

For a proof of this theorem, we refer to Royden [184], and Gardiner [A-34],Theorem 5 in Chapter 9.

Proof of the surjectiaity o/i*. Now, we return to a proof of the surjectivity of f..

Frorn the previous observation and Theorem 6.22, for every / e ,Aut("(f)) and

every point p e fQ) there exists an element ["0] e Mod(f) with [c..'o]-(p) =

/(p). W" need to show that [c.,o] can be chosen independently of p. Fix a point

q e TQ) arbitrarily. Recall that "(l-) is biholomorphic to a bounded domain

(Theorem 6.6), that the Teichmiil ler distance d is complete (Theorem 5.4), and

that Mod(f) acts properly discontinuously on "(l-) (Theorem 6.18). Then we

can find a positive constant 6 so that d(p,[r].(p)) > 26 for any p € "(i-) with

d(q,p) ( 6, and for any lule Mod(f) with ["]-(p) lp. Thus we have

'([69] uralsdg 3c)crdolosr are feql;r fluo pue;r crdolouroq ers eceJJns pesop e yo s3urddetu o,lr1}eql l?eJ u^rou{-lla^{ ,{11ecrsse1c eql ll€car osl€ aA '*u fq l-r elouep pu€ 'ar sr a?eJ-rns Surr(1.repun asoqAr aceJrns uuetuerg e xg e^\ '.raq1.rng 'Surarasard-uolleluauo

pue crqd.rouroauoq sr Eurddeur fra,ra 1eq1 pu€ '(Z ?) 6 snueE Jo er€Jrns ("lq*lf

-ua.ragrp) pesolc peluerro rre q U ?eql etunsse sfeule aa,r 'uorlcas sH? uI'([tSZ] uolsrnql pu€ gg'g ruaroeqtr aes) ace;rns pesop e;o s3urddeur

-JIes Jo rueroeql uorlecurssel, s(uolsJnr.{tr-uaslarN el€q er* 'acuenbesuoc e sV'[rzt]"rx puts'[28-V] uaslerN'[Ot-V] ralralg pue uoss'eC ot raJer osle e11\'[OZ-V]nr€ueod pue qoequapnel 'yq1eg aas 'sluatuleerl

IInJ rog 'adf1 elrug {leerldpue

Jo eceJrns (pasolc fl.ressaceu 1ou) e;o ese) eql ur paraprsuoc oqe sr uorlersrsselce qcns '6 snua3 Jo e?"Jrns pasoll e go dnor3 sselc Surdde- "qt Jo sluetuele Jouollc€ aq1 'f11uap,rrnba ro '(Z

< 5) t; ;o suorleruroJsueJ? relnporu railnuqcretr

Jo uorlecgrssel? Jelnurs " ssnc$p II€qs eAr '[gg] srag Surr',ro11o;'uorlcas qq] uI'rqoqered pue 'crloqradfq 'cr1dq1a pelle) ere qrrq^,r

'sad,t1 eerql olur 6 raldeqC ut pegrsselc ueeq elerl suorleuroJsuerl snrqotr{ IeeU

suorlBrrrJoJsue.I,I,

rBInPotr I rallnuqr.ral Jo uorl€rgrss€lc '9'9

'p1otruoru uNals D s? (ilJ acods .ralputp?eJ eqJ 'e7'g rrraroaql

'ueJoaql Surr'ro11o; eql e^Bq e,.l\ aJueg',,tqdrouroloq

Jo ursuop e s1 (.7)ag leql luep^rnba sr srql '(trtt 'A '67 uraroeq;'[rt-V] srag) ura.roeq] s,e{O rg '(1, .ra1deq3 ut ?'t tuaroaql'[tq-Y] rqsefeqoyeas) xa,ruocopnasd sl (.f)s't snql 'ecu€lsrp rqse{eqoy eq} o1 lcadsa.r q1r,rlelelduroc sl (J)sJ lerll epnlcuoc airn '16'9 pue l'g $uaroeqtr uroq 'fleurg

'02'g ureroeq; pa,rord el€q e,lr snqJ'(J)J uo rf - *['o]

1eq1 sarldrur suorlrunJ crqdrouroloq roJ ruaroeql ssauanbrun aql 'pelleuuoc s-r(.i,)g acurg 'g > (d'b)p qlt,r (J)J f d 1p .rog (d)/ = (d)'fdnJ - (d).[to] ''e'1'il = (d),ldnlor'[to] ]€qt s.trolloJ 1r 'ecua11 'g > (d'b)p qll,r (J)J ) d 1e .roy

9z > @'b)pz =

(@)l'(l)t)p * (b'a)p -

((d) - [o"4' (D). [or] )p + ((r) -[,r'l]' (d),lo r))p ]( (d) . [dr]' (d).lo rl)p = ( (d) . [do] o, lfb ol' a\p

suorl"urroJsuPrJ r"lnPoIAI re[nuqf,reJ Jo uorl?f,ursselc'9'9 TLT

172 6. Complex Analytic Theory of Teichmfiller Spaces

6.5.1. f\rndarnental Extremal Problems

We can deal with the classification of real Mobius tra.nsformations relating to

an extremal problem on hyperbolic translation length. More precisely, for every

element 1 e PSL(2,R), set

a(t) = i \ f_nQ,tk)),

where .E[ is the upper half-plane, and p is the Poincard dista.nce on ly'. Then real

Mobius transformations 7 are classified as follows:

(i) 7 is elliptic if o(7) = 0 and there exists a point zt

PQt,l@t)), i 'e', z, is a fixed Point of 7,(ii) 7 is parabolic if o(7) = 0 but there exist no points z,l

€ I/ with a(t) =

€ I/ with a(t) =

p(zr,yQr)), ar.d(iii) 7 is hyperbolic if a(7) > 0 (and then, there always exists a point z, € H

wi th o(7) = p(zr ,yQt)) ) .

Now, for Teichmiiller modular transformations, we consider the following

similar extremal problem.Bers' extremal problem for Teichmiiller modular transformations.

For every Teichmiiller modular transformation X of ?(R-)' we set

a(x) = o.#it".,

d(p,x(d)'

where d is the Teichmiiller distance on ?(.R-). Then find a point px € T{n.)

such thata(x) = d(p* ,x@)) .

If there exists a solution p, e T(R*), then we call p* a y'minimal point- We

classify Teichmiiller modular transformations 1 into four types:

(i) 1 is elliptic if o(x) = 0 and there exists a x-minimal point (which should be

a fixed point ofX),(ii) 1 is parabolic if a(x) = 0 but there exist no x-minimal points,

(iii) X is hyperbolic if o(1) ) 0 and there exists ax-minimal point, and

(iu) X is pseudo-hyperbolic if o(1) > 0 but there exist no 1-minimal points.

Note that this classification is independent of the choice of the complex struc-

ture on .R which is used to define the Teichmiiller space T(R-)'

Recall that every point [,S, /] € "(n-) is represented by [Ro, fd] (see $1.4.1 of

chapter 1), where Ro is a Riemann surface equipped with a complex structure a

on fi, and rd is the identity mapping of .R* onto fto. Hereafter, lRo, f dl is simply

written * ["].Every Teichmiiller modular transformation is represented by [f]*for a self-mapping "f of .R (see $3.1 of Chapter 6).

Now, to investigate x-minimal points, we may consider the following version

of this extremal problem.

Berst extremal problem for complex structures. For every complex

structure o on .R and every self-mapping f of R, considering .f us a self-mapping

ursrqdroruolne atqdrouroloqlq e sl / ?Bql qcns X Surcnpur A Jo { Surddeur-g1ase pue Ar uo , ernl?nr1s xalduroe e sr ereql uaql '1urod pexg e seq x JI

'loo.t4

'ctpoutail s? t! l?fr.1uo puo 11 cpTdgla st X uorTouttotsuo.tT rnlnpout reIInurUcNU v

'g7'g tuarooq;,

' u^,i,oul-lla^,r f gecrs-s"lt sr uaroaql Sura'ro11o; aq1 'suorleruroJsuerl relnpour rallnulqereJ cr1d11e rog

suorleruroJsrrBr,l cnoqradfll pue arldwlg'Z'g'g

tr 'uorlresse aqt ePnleuol ein Pue 'fFe[urrs ua{oqs st asra^uoc eqJ

'1eu1u1tu-*[rf] "l oa'acue11

.((td).[/] 'rd)p ] ((od).vl,od)pa^"q

aal'acuelsrp rellnurqrral aql ol laadser q1/rr (-U)Jgo frleuosr ue sr *[3f] ecurg

'(( rd), :[/]' td)p > ((od), j[/]' od)p

o1 luap,rmbasl (UI'g) flqenbaur'I'e$ u! (,A)l uo rf go uortre erllJo uoltrugep eqt,{q ueqA'flaarlcedsar'Io pue o o1 Surpuodsarroc (.U),2 ur slurod eql aq Id pue od p.I

(zrg)'(l)'"x ) Pl)"xeler{ ein 'g uo Io arnlcnrls xelduroc i(re,re pue / o1 ordolouroq U Jo V

Surddeur-g1as.{rana.ro;'uorlrugep eql ,ig '(6'9 ruaroeqJ lc) ? aceJrns uueuergaqt uo / o1 ordolouroq (Surddeur lerueJlxa anbrun aql ''a'r) Surddeur reilnuqrral

aql eq oI p1 'leununu-;| sru arnlf,nrls xeldruoc e l€rlt asoddng 'loo.t4

'lorututra-'fl) st o o7 |utpuodseuo? ("A)J ) [o] Tutod aq1 lt fi1uo puo tt louttutul-l s! o ernl?nrls xaldutoc o'A lo t 6utdtlou.t-11?s D rof, '?Z.g uolllsodo.r4

'uol?resse Surr*o11o; aql a^"q e,rl ueqJ:(6'9 ureroaqJ Jc) ooy * o"A , t o1 ardolouroq eJe qorq^,r ooy * ooq :rt s3urddetu leur.ro;uocrsenb

ge ;o flurey aql ur Surddeur leueJlxe anbrun aq1 ''a'r 'Surddeur rellnuq?rele q o,U * ooU:o/

terl? palou $ ?I'o"U a?eJrns uueruarll arll Jo |ut,dilpru-t1as pua.tTra fi1a7n1osqo ue org * ooy:o/ pue ernllnrls talihuoc lou.uutut-tue 0p IIec e,rl uaql(paqr.rcsap s" (0/'0o) r-red e 'uor1n1os " slstxa areql JI

'{ otctdolouroq A p rl Surddeu-gas f.rarla pue gr uo r, ernlcn.rls xalduroc {.rerr.a .ro;

(l)'"x > (t)'"x

l€ql r{cns / o1 crdolotuorl U Jo o/ Eurddeur-;las e pue U uo 0, ernlcnrlsxaldtuoc € pug uaqJ 'leuJoJuocrssnb lou sr / lt * = U)"X 1nd aal 'ara11'/ go uotlelellp I€rulxeur eW $)'>I {q elouap ar\{ ',U ec€Jrns uu€urenl aq} Jo

t/Isuorl" ruroJsu"rJ r"ln PoI,\l rallnurqf, ral Jo uorl"f, ursselc' 9' 9

t74 6. Complex Analytic Theory of Teichmiller Spaces

of Ro. Since ft is compact, it is well-known that / should be of finite order (see

Rema.rk 2 in $6.3), and hence is periodic.

Conversely, suppose that X is periodic. Nielsen showed that x has a fixed

point in ?(E-), whose proof we shall omit here. (Actually, it is shown that the

action of every finite subgroup of Mod(R*) has a fixed point in "(,R-). This is the

affirmative solution for Nielsen's realization problem (cf. Notes of this chapter).

See for instance Kerckhoff [112] or Wolpert [256].) D

Remark. A weaker version of Theorem 6.25 is easily shown. Namely, it is easy

to prove that a self-mapping f of R is homotopic to a periodic self-mapping of

-R if and only if there exists a complex structure o on R, and a self-mapping /shomotopic to / such that /e is holomorphic on .Ro.

Note that Proposition 6.24 implies the following theorem.

Theorem 6.26. Let f be a self-mapping of R. Then lhere is an f-rninimalcompler stracture if and only if the Teichmtiller modular transformation lfl.corresponil ing to f is either ell iplic or hyperbolic.

Now, a finite non-empty set {Ct, " ' ,Cn} of mutually disjoint simple closed

curves on .R is called admissible if every Ci is freely homotopic to none of

{Cx,(Co)-t}*1i, and is not homotopic to a point. We say that a self-mapping

f of Ris reduced bv {G,"' ,Cn} if this set is admissible and

f ( C t u . . . u C " ) = C r U . - . u C n .

A selfmapping / of r? is called red,ucible if it is homotopic to a reduced mapping,

and. irceduciDle if not. Then we have the following theorem.

Theorem 6.27. If f is an irteducible self-mapping of R, thenthe Teichmiil ler

moilular transformation [f]. induced bg f is eilher elliptic or hyperbolic.

To prove this theorem, we prepare several lemmas. First, we start with the

following fundamental one.

Lernma 6.28. (Wolpeft) Let f be a quasiconformal mapping of a Riemann

surface 51 onto anolher 52, antl C be a simple closed, geodesic on 51 with hyper-

bolic length \. Then f (c) is freelg homotopic to a closed geodesic uith lenglh

lc such thal

t 2 < K ( f )h ,

where. K(f) is the maximal dilalation of f .

For the proof, see that of Lemma 3.1 in Wolpett [2aG)'

(6 .13)

Next, by the collar lemma (cf. Matelski [150]), we can easily show the follow-

ing:

1eq1 satldtut 0g'g etutuerl 'acua11 'elqrcnparrr osle $ {t1 d.rarra ,elqrcnpe.r.rr sr 3facurg 'f i{ra,re .ro; V > (0>t }€ql q)ns y luelsuoc € sr areq? ,.re1ncr1red u1

'(.[/]), = (t,ixao1ffifeleq eal '(gt'g) dg

Eurddetu rellnuqcral aql eq ooq - t"A , !r! lel pue ,(!d)-Vl

turpuodsarroc ern?f,nrls xalduroc eql aq lo pI , [ ,{re.l,a .rog

'(.[/]), = ((td).[/] 'ta)pTir1!

';l' o1 erdolouroqpue ld qloq ot

(qrg)

leqt qtns (.U),2 q r;i{fd} acuanbese e{eJ 'La'g ueroeqJ {o !oo.r4

'(.a)l ut' sa6.t'aauoc r-;f�{("!d)ux} acuanbas?T? tDrg qcns (.9)alo suotTont.totsuorl rvlnpou rellnuy?reJ lo t--f.{"X} acuanb-es D puD 't=j{fd}

lo r-*{"!d} acuanbasqns o slsrce ereql ueqJ 'g uDrll .ra7oa.rosx fg qcoa uo crcepoa| pasolc alduts fiuo lo r176ua1 cqoqtadfr,tl aUl ?Dql Vxnsg aatpsod D s, ?raql 7oq7 asoddng 't fi.taaa .r,ol {fl o7 |utpuodsa.u,oc aco!.tnsuuDurety aq7 aq lg puD '(,A)J ur ecuanbas , ,q t{{!d} ?aI .Ig.9 EurrrraT

'([69] srag e?u€tsurrog 'yc) ueroaql sseulcedruoc s.proJirunq 3uralo11o; eq? ilecer arrr ,f11eurg

tr 'uorldurnsse eq? sl?rperluoc q?rqlr 'alqranpe.r

eq plnoqs / snqJ,'(s1as slurod s) 0, = ("C),1 pue (I - r' ...t0 - 9) r+!g

= (C),1 teql qcns ,/ Surddeur e o1 flsnonurluoc 3f urroJep u€r a..r{(ra,rau,o11'(s1as slurod se) 0, - t|'C pue lurolsrp e.re

'C' . . .'0, leql qrns r re3alur alrle8eu-uou e sr eraql 1"ql 6U.g €rrruarl ,{q aurnsse.{eur a,u '9'g uorlrsodor4 {q lugofqp er€ tueql Jo lle ?ou erurs .0g ueq} ssal sr

e-rEC' ...'oCJo euo.r(ueyo {fua1cr1oq.red.{q aql ,(gI.g) fg .(g -6t, ...,I = [)(C) { "l crdolouoq f1ea.g crsepoe8 pasolc alduns eql eq fg 1"1 p,rn , = 0, tas

'og > i. '-oe?))t''n'l 'plo.I lou seop (p1'g) l(lqenbeur '3[ euros JoJ '1eq1 esoddng'i�oo.t4

(rrg)

DurueI ux ?uD?sun ay7 st 09 a.taqm

? (/):rsa{sz7os S lo I

6uzddout-t7as elqr?nperr, fi.taaa uaqA'k7) 0 snuaf lo g acnl.tns uulurexq pesopn uo V r170ua1 ctloq.tadfiq y.tn nsapoa| paso1c aldtuts D ?q C leT .Og.g eururaT

:turrr,r,o11og eqt a^eq ar'r snqtr

'09 uotg ssel ero uaqT lo st176ua1 ctloq.tadfiy IDW pepraord Tutotstp a"tp 6 snua|

[o acot.tns uuDur?ry p?soli D uo sctsapoaf paso1c alduts l?ut?srp omy fiuo 7ot17qcns'6 snua6 uo fr.1uo spuadap Wlqn'0 <0gluDlsun D s, at?qJ.6Z.9 eururarl

'64'9

,r-rrrrr(u?)

9LIsuorl"urrolsusrJ r"lnpow raflnulrlf,ral Jo uorl")yrss"lc . g' g

L76 6. Complex Analytic Theory of Teichmiiller Spaces

the hyperbolic length of any simple closed geodesic on each Eo, is greater than

6sA3-3t. By Lemma 6.31, we may a^ssume, taking a subsequence if necessary,

that there is a sequenc" {Xi}Er of Teichmiiller modular transformations such

that the sequence {Xr(pi)}Et converges to a point q € ?(R-). We set q1 =

Xi@) for every j. Since eaih Xi is an isometry with respect to the Teichmiiller

metric, (6.15) gives

ilg d(qi ,Xi olfl* o (xi)-'(ci)) = o([f].)' (6 .16)

Again taking a subsequence if necessary, we may a^ssume that {xi o [/]- ?(Xi)-t(qr))Fl converges to a point q' eT(R,), for ?(r?.) is finite-dimensional

and is complete with respect to d. since each 1r. o [.f]- o (xi)-' is an isometry, it

is easy to see that

itg xi o [.f]. " (x.i)-'(q) = c''

Hence, by Theorem 6.18 we may assume, taking a subsequence if necessary, that

Xi "lf).o (Xr.)-1 has the same action on "(ft.) for every sufficiently large j, say

j > j o .

Then (6.16) implies that

d(q,x ioo [ / ] - o (xr" ) - t (q)) = d((x i " ) - t (c) , [ / ] . ( (x i " ) - ' (q)) ) = o( [ / ] . ) '

Thus there is an [/].-minimal point, which shows the assertion. n

6.5.3. Absolutely Extremal Mappings

Next, we cha.racterize absolutely extremal mappings. We have seen that, when

a self-mapping / is homotopic to a periodic mapping, then the corresponding

absolutely extremal mapping is conformal (Theorem 6.25). Hence, we discuss

the case that [/]- is of infinite order.

Here, note that the Teichmiiller space ?(fi-) is a straight line space in the

sense of Busema,nn (cf. Kravetz [128], and also see Masur [142]). In pa,rticular,

any two distinct points p|,pz e "(ft-) lie on a unique stroight line, say tr' which

is an isometric image of R into "(E') equipped with the Teichmiiller metric,

and contains all points p such that

d(pr, p) * d(P, Pz) = d(Pr, m) -

We also note the following elementary fact:

Theorem 6.32 If a Teichmiiller moilular transformation 7 is of infinite oxler,

lhen a point p € "(R,) is y-minirnal if and only if 7 leaaes a straight line thrvugh

p inaoriant.

Proof. Assume that p is x-minimal. since x is of infinite order, three points

p, X(p), and X2(p) are distinct. Let p1 and, p2 be the midpoints of the "seg-

-"tttr'; [p,x@\ and [1(p), xz(p)], respectively. Then it is ea'sy to see that

1eq1 saqdurl (gt'g) €lnuroJ 'GI)X ! (V)y a?urs pue ,(U) Ot.l rueroaql

fq zU)X ) ("t)X e?urs 'leurer1xe f1a1n1osq€ sr / feqt etunsse ,alolg

(srs)

'./ o1 ctdolouroq Surddeur Jallnuqcretr eql $ 11 ereq^r

,(u)>t = "U)yse uellrrr,r,er q (ft'g) 'Surddeur rallnurqrreJ e sr 3l aaurg

(rrg)'( (["]).(.[/])' [r])p =

((["]).[/] '[o|)pz =((["])'(. Ul )' (["]) - [/] )p + (([o]).[/]' [o] )p

o1 luele,rrnbq fl s-rrll ,69.9 ureroeq;, ,tg .leurut.tu--[/] sl(.U),2 ) [o] lutod Surpuodsa.rroe eq] JI {po pue 3r Isuer}xe flalnlosqe q !

'V?,.g

uotlrsodor4 fg 'o ernlcnrls leuroJuoc elqqlrns e rlty( oA = S 1n4 .Eurddew

rallnuqcratr e sr erueq pue 'leuro;uoc lou q / 1tsq1 erunss? Aern a71y .loo.t4

'z$)x = (z|)x qTtm |ut'ddotu rennuruz'eJD oslD cr 7t futrldou otp to1? qcns |utildour rqpuq)r4 o ro |utddotu Totu^tot-uol D r?qpe s, 7t lt fryuo puv fi lourar?se fr1ayn1osqn s! ,g - g : I 6ut,ddou.t, oueqJ '(?,

-) 6 snua| to acottns uuouery p?sop o aq g pT .gg.g uraroatll

'rualoeq? Eur,raollo; eql ur"lqo eiu 'alo11

'[167] ealie?ru€tr osp ees 'auq l{Ere.r1s ?uelr?Aur auo ?soru +e s€q uorl"uroJ

-su3r1 relnpolu reilnuq?ral /tue l"ql ,(roaq1 s(uolsrnqr ruog s^rolloJ lr'I clrDuea

'luougau, autl 7t16to.r.7s D setuDel 7g tg fr1uo puo tt u1oqtadfiy sg *flf uo4out^totsuo.tTrDlnpout rellnuq?pJ eW'(A)poW > fl)Wauala atpouad-uou D ro4 .r(.re11o.rog

O '1eururur-X s1 d pqt sarTdurr qclqa'(-U)J) d f.ra,ra ro;

((d)x'd)p 2 ((d)x'd)p

leqt apnl?uoc eal 'f.rerlrqJe sr u a?urs

' (( d)X' d)p . u * ( d' d)pz > ((d) "x, d)p = ((d)x, d)p . u

teqt'flllenbeut a1Euet.r1aq1 Eursn fq 'aas uec a^\ 'u reSalur a,rrlrsod fue pue (?),2 3 d lurod fue .ro;'ueqA 'X rapun luerr€Aur sr d qEnorql Z eull lq3rerls € lsql asoddns ,1xap

'X .rapun luerrslursr 'eU ((d).X (ed 'rd osp eruaq pue) (d)X ptre d qarq,u uo ,aur1

lq8rerls aql leqtsarldurr qcrq^a([zd(rd] ?ueur3as aq1 go lurodplu eq] sr (d)X snqa'(X)oJo uorlru-gep aqt fq (X)o 7 (za'41, eAeq a^{ '(a)X - zd aculs ,pueq .raq1o eq} uO

'(x)' j (zd'rd)P leql epnlcuoc aiu' 'acua11

.(x)"1= (d,(d)x)p = ((d)x,td)p

suorl"ruroJsu"rJ r"lnpon railnuqtral Jo uorl"f,urss"Ic .g. g LLI

178 6. Complex Analytic Theory of Teichmiiller Spaces

I{(h) = K(f,).Hence, the mapping .f2 is also a Teichmiil ler mapping with

xift)= I{(-ff . (Note that, by Theorem 5.9, we conclude that h = f2.)

Conversely, if the mapping /2 is also a Teichmiiller mapping with K(/2) =

K(f)2, then clearly (6.17) holds. Therefore, / is absolutely extremal. E

Remark 2. we can show further that the condition that the mapping f2 is also

a Teichmiiller mapping with 1{(/2) = K(f)2 is equivalent to the condition that

the initial and the terminal differential of / (cf. Proposition 5.19) coincide with

each other up to a positive constant factor.

6.5.4. Reducible Mappings and Nielsen-Thurston's Theorem

For reducible mappings, we can show the following theorem.

Theorem 6.34. Let f be a rcducible self-rnapping of R. If f is nol homolopic

to a periorl ic mapping, then lf l. is either parabolic or pseudo-hyperbolic.

Thus, by Theorems 6.25,6.26,6.27, and 6.34, we conclude the following:

Corollary. Let f be a self-mapping of R. An f -rninimal complex structure exists

if and only if f is either homotopic to a periodic mapping or irteducible.

We shall omit the proof of Theorem 6.34. Instead, we shall explain the struc-

ture of a reducible mapping. (For this purpose, recall that the foregoing argu-

ments still work even for the case of a surface of finite type.)

Let / be a reducible self-mapping f of R. Then we can deform / continuously

to a completely reduced mapping, or more precisely, to a self-mapping fs of ft

which sa[isfies the following condition: there is an admissible set {C1,... ,Cn}

of disjoint simple closed curves on R such that, for every comp-onent H of R -

CtU-...UCn a.nd for the smallest positive integer N with /Jv('?') = R', the

mapping /t l,*, it irreducible. \

Let { .Ri , . . . ,R;} be the components of R- CrU" 'UC,. As in the case of a

closed surface, there is "r, (/", Ini )-minimrl complex structure oi on Rli and an

absolutely extremal mapping Fit (H)", - (R!)"1for every i ( i = 1,"' ,m),

where Ni is the smallest positive integer such that |Ni(Rii) -- Rli (We can

further slow that (Hi),, is a Riemann surface of analytically finite type.)

We can show that

o ( [ / ] - ) - max { I i o , ( f ' 1 ) , ' ' ' , Ko^ (F^ ) } ,

but there exists no [/]--minimal point.

Hence, if all K"r(S) are equal to 1, or equivalently, if all .Q are conformal,

then [/]- is paraboiic. If not, then [f]. is pseudohyperbolic.

'([t t t]Uoq{rrey pue [6S] srag 3r) [911] uolsrnqJ pue Joq{rrex ees 'auo s(uotsrnt{JruorJ lueJeJrp sr Surppaqure (srag Sursn t

6 Jo uorlergrlceduroc aql leql alou oslee \ '[qql] uefln4ctr{ pue '[8gI] tl{sel tr '[96] srag '[t]goUqV o] raJer alll'1ca[qnssrqt rod 'sdnor3 u€ruraly ;o ,froaql aql o1 palelar .{1aso1c q (.f)sz Jo (J)sJgfrepunoq eql'0 <u*Z-OZWql pepnord u+t-6e) ul (J)aJ uretuop papunoqe se (.7)g sezrlear Surppeque ,srag

'relncrlred u1 'f snua3 Jo eceJrns uuetuelgpesolc € ruorJ slurod l)urlsrp u Surrlorue.r fq paurelqo sr r{3rq/r\ a?eJJns uu€ruelge ''a'l'(u'f) adfl alrug d1ecr1,(1eue;o s\ JIH ec€Jrns uueruelg aq1 ;t dluopus JI I€uorsueurp alrug il (.f ),_f t"qt u^,rou{ sl }I 'J dnor3 uersqcng ,trerltqreue 1o (.7)"6 aceds .rellnruqclel eql o1 alqecrldd€ sr poqleu slqJ'[qU] srag o]enp sr raldeqc qql ul r; go arnlcn.rls xalduoc aql eugep ol poqleur eq1,

'([ogt] u{sew pue [gul] ery aas) serDurpron pqsDw eqr

pelF) ar€ qclq,h (J)J uo salsurprooc crqd.rouroloq lrcrldxa JeqFJ a^erl eA\'[ggt] ott"S pus'[?gI] ,(1s.r'o[ra1 pue Eeg '[g1] equro"rl

pue raqrsrd '[Of-V] lsof ur punoJ er€ 1calqns slr{l Jo s}uerul€arl e^rleurelly

'y xrpuaddy ur fgar.rq peureldxe eq [eqs se'[g] sroJlqy.,(q pecnporlur lsrg*^ (U a) f snua3 3o

tg aceds rellnuqcrel eql Jo arnlcnrls xalduoc aqa'ltt-vl €rlspox pue ^rorrotr l pu€

'[fS-V]€rrepoy '[Og-V] srrreg pu€ srltglrg '[ft-V] s.rag ,tq qooq eqt ot raJar a,lr'splogrueu xalduroc pu€ selq€rrel xalduoc I€JaAes uo slerJeleru l€tue{uspunJ rod

saloN

'(yloq \ou 1nq) alqtcnpa.t st .r'o uL,sttld.r,ouoa[r,p aosouy-opnasd

o o7 ct,doToutoy st'auo ctpouad o o7 ctdoTotuoU pu s, q?tqn'(27) 6 snua| toacottns pesop o to futildoru-l1as y (uolsrnql pue uaslarp) 'gg'g tuaroaq;,

'uolsrnql pue ueslerN Jo rueroaql Surrrrollog eq1 fldtur ?g'g pu€ /6'9 suraroeqlsnqJ 'rrloqradfq sr -[/] yr fluo pue yr usrqdrouoeJrp ^osouv-opnasd e o1 crdol-ouoq sl / leql fldurr g'g$ ur A {reru€rg pu€ 'tt'g '96'9 sure.roaqa 'e.royeraq;

'rolceJ lue?suo,allltsod e o1 dn raqlo q)se ql-ri apIDuIo? slelluareJlp leurturel pue I€IlIuI esol1\^Sutddeur rallnuq)retr € seurooeq / tn,It qcns U uo arnlf,nrls xalduroc € ulelqou€? a^! 'A

Jo { tusrqdrotuoegrp ^osouy-opnasd ua,rrE due ro; '.{1esre,tuo3

'ernlcnrls xalduoc eq1 ?a3.ro3 a r uarlirr 'u.tsrtld,totuoaficp aosouv-opnesd pell€c-os e se,rr3 rolceJ luelsuoc errrlrsod e o1 dn raqlo qc"a qllrrr eplc-uroc slsrluaraJrp l€unuJal pue Ierlrur esoq^\ Surddeur rellnurqcrel e '.{1eutg

'cqoqered

sl -U] 1"ql ees uec aal 'acua11 'C - A ;o luauodtuoc qcee uo Surddeur .r(1r1uepreql ol crdolouroq sr / pue 'asro.luauodtuoc

lu€Irelur C - A ser'ee1 / ueqtr, 'C

a rnc pesolc aldturs e o1 lcedsar qlr^r lsrlrl uqeq eql ag U - A : I p1 'aldutotg

6LIsaloN

180 6. Complex Analytic Theory of Teichmriller Spaces

Let "s(1) be the image of Bers' embedding of the universal Teichmiillerspace ?(1), where I denotes the trivial group. Gehring [83] proved that "3(1)

coincides with the interior of the set S(1) consisting of Schwarzian derivatives of

holomorphic univalent functions on 11. It is also known that S(1) S "s@ (see

Astala [19], Gehring [84], and Thurston [232]). Shiga [200] showed that if l- is

a finitely generated Fbchsia.n group of the first kind, then fBQ) coincides with

the interior S(l-), where S(l-) = S(1) n Az(H*,l-) ' When f is of the second

kind, Sugawa l2,l2l has shown recently that S(l-) I fn!).However, in the case

where l- is infinitely generated and of the first kind, it is unknown whether

S(f) c ra!) or not.

For an arbitrary Fuchsian group i-, the Nehari-Kraus lemma (Lemma 6.7)

implies that "s(l-) is contained in the open ball with center 0 and radius 3l2in

Az(H., l-). The infimum of radii of open balls with center 0 in A2(H* ,i-) which

includes fBQ) is studied by Nakanishi [166], Sekigawa [192], and Sekigawa and

Yamamoto [193].For connections with projective structures on Riemann surfaces, there are

papers Gunning [88], Kra [120], and Shiga [202].

It was known by Fricke that the Teichmiiller modular group Mod(l-) induces

a discrete subgroup of Aut(T(f)) (see, for example, Fricke and Klein [A-33]).The proof in this chapter is due to Bers [31]. The modular group Mod(l-) is

studied in Bers [39], Hejhal [100], Ivanov [108], Kerckhoff [111], McMullen [154],Mumford [161], and wolpert [249]. For classification theory of Teichmiiller mod-

ular transformations, we refer to Casson and Bleiler [A-19]' Fathi, Laudenbach

and Podnaru [A-29], Bers [38], Kra [121], Shiga [201], and Thurston [231]. Kerck-

hotr [112] and Wolpert [256] solved the Nielsen realization problem which asserts

that the action of every finite subgroup of Mod(l-) has a fixed point in ?(f).

See also Kerckhoff [113].For Bers' fiber space over a Teichmiiller space and the Teichmiiller curve,

we refer to Bers [31] and Earle [61]. The relation between Teichmiiller spaces

and holomorphic families of Riemann surfaces is found in Nag [A-80]' Chapter

5, Earle [58], Ea"rle and Fowler [64], and Imayoshi [102]. Their applications are

treated in Griffiths [87], Imayoshi [103], [105], [106], Imayoshi and Shiga [107],and Riera [183]. For holomorphic sections over a holomorphic family of Reimann

surfaces, see Hubbard lA-441, and Earle and Kra [66].Many more details of the Kobayashi distance a.re found in Kobayashi [A-

54], Lang [A-64], and Noguchi and Ochiai [A-88]. For a distance on ?(,l-) in-

variant under the action of. Mod(f) other than the Teichmiiller distance, we

have the Carath6odory distance. Earle [59] proved that "(l') is complete with

respect to the Carath6odory distance. The connection between the Teichmiiller

and Carath6odory distances is studied in Kra [122]. There are also invariant

distances which are induced by the Bergman metric and the invariant metric

on the Siegel upper half-space. For this subject, see Royden [185]. Even in the

case where "(l-) is of infinite dimension, it is shown by Gardiner [81] that the

Teichmiiller distance also coincides with the Kobayashi distance. See also the

book by Gardiner [A-34].

'rrrell Jo lurod srql urorJ palPnts st se?eJrns uuetuelg ;o t(pue; e ;o

uorprrel Isrursalruuw aql eraqa '[996] lradlol\ ol reJar osle eA\ '1 re1deq3lo 7'6$ur fgarrq pessncslp eq ll€tls sploJlueur xelduroc ;o f.roeql uollsturoJeP .racued5-eiltspoy aql pu" saceds relpruqcle;, ;o ,(.roaq1 eql uaa^r}eq uolleler eq&

.[996] r.radlolt pue [66I]e3rqg ur s;oord e^Ilsurelle errl areql 'ploJlueur ulals € st aceds railnuqoleJ

I"uorsueurp alruS e t"ql h?] srarduarqg Pue srag fq pe,rord 1srg se^{ 1I

I8I

Chapter 7

Weil-Petersson Metric

Unless otherwise stated, a Fuchsian group l-, considered in this chapter, is aFuchsian model of a closed Riemann surface of genus g (Z 2).We also assumethat each of 0, 1, and oo is fixed by a"n element in f - lid\.

As stated in $1.3 of Chapter 5, the Teichmiiller distance on the Teichmiillerspace 7(.l-) measures a kind of magnitude of deformation of complex structuresof Riemann surfaces, and ?(f) is complete with respect to the Teichmiillerdistance. We also saw in $3.4 of Chapter 6 that the Teichmiiller distance is equalto the Kobayashi distance, which is defined complex analytically. However, theFinsler metric induced by the Teichmiiller distance is not of class C-.

The purpose of this chapter is to introduce another natural metric on ?(l-),which is called the Weil-Petersson metric, and to prove that it is a Kd,hler metricwhose Ricci curvatures, scalar curvature, and holomorphic sectional curvaturesare negative.

The first section is preliminary and devoted to studying the Petersson scalarproduct on the space ,42(.I/, f) of holomorphic automorphic forms of weight -4

with respect to l- on .Il, and related topics such as the reproducing formulafor holomorphic automorphic forms, Poincar6 series, and the Bergman projec-tion. In Section 2, we see thaL A2(H,,l-) is the dual space of the holomorphictangent space Zs(?(f)) of TQ) at the base point. We also represent elementsof "r(Z(f)) by harmonic Beltrami differentials. Then the Weil-Petersson met-ric on "0("(f)) is given by the dual metric of the Petersson scalar producton A2(H,l-). In Section 3, we define the Weil-Petersson metric on 7(f) andverify that it is Kiihlerian and that its Ricci curvatures, scalar curvature, andholomorphic sectional curvatures are negative.

.udtv(udt,dr\7 - d, ",.\",-,/J_-,

urroJ eql uI uelllr^{ sl (V)zV I d flera 'relnrtlred q'(y)zy to1srseq leurouoql.ro elalduoc e sI 0--J{"6 } t"qJ'u ra8alut eltleSeu-uou.{ue.ro;

(r'z) ,z (t, + u)(z +uXr + q: l\ - G)"d3

1nd a111'lcnpord rel"cs srql q1r,n eceds treqllg alqe.redes e sauro?eq (y).y t"qf

npxp (z)rlt(z)d) z-e)y" [ [ = v (,/r' 6)JJ

fq(V)zV uo v( . '. ) tcnpord rel€cs uossrated eql augep eM'y {slp llun

aql uo clrlatu erecurod eq1 * "lzplr(")V = zsP ''a'\'("ltl - t)lZ = (z)y areqm

'a > npxp ,l?)al "-Q)u"l[ = )l^ttl"ql qrns

V uo d suorllunJ crqd.rouroloq ;o areds roltel xelduror eql aq (V)eV +"1*;. eueld-;pq

remol aqt uo J ol lcedse.r qll^r sruroJ crqd.rourolne ctqdrouoloq lo (J'.H)zVuo lcnpord relecs uossJeled eql eugep eAl'r(errr aures aql uI '(tJ'�V)eV pue

(l'n)"V seceds lraqllg rltoe esn eA\'(/J'V)zV uo lrnpo.td rel€?s uossreledeq1 'flrepurrs eugap e$'V {srp ?run eql uo 3ut1ce E Io il Iepou uelsrlcnd €

3uqe1 'g aueld-;1eq raddn eql uo 3ur1ce U Jo J Iapou u"Isq?nJ B Jo peelsul'(l'n)zV uo 7cnpo.r.d rDlms uossreled aql u(''.) lcnpo.rd reuur uellpurell slt{l

II€r el!\ '1cnpo.rd Jeuul uelllureH slql qtyr,r aceds treqllH e setuoceq (l'n)zV

l€rll s/rtolloJ 1r 'lceduroe $ u atuls '.H ul J roJ ulsruoP lelueuepunJ e sI Jr eJaI{^{

Jf f Uf f'fipxp(z)rfu(z)dtr-(z)xu

ll = op(,/L'd)

ll =a(,/r'6l'

fq paugap q u({1 'd) lcnpord reuur uer}runa11 aq1 'r*o11

ar uo uorlcunJe s€ pereprsuoc sI 1l

''a'-I '.y rapun lu?lrelul l? uo uolltunJ e sl (41 'ai) uaqS

'H ) z '(z)rfu(z)duly =(z)(r/t,'dt)

les e a '(J'H)"V ) ,lr'o\ sluetuela due rog'J

/n = Ur a?"Jrns ulrctuarll aql uo luetuala€are ue se papre3a.r s\ lsp fq pacnpur frpap"(z)ny - ,p luaruele eere eqJ,'r_("*t) -

@)nU ''e'\'H uo rrrleur ar€rurod aqt aq "lrplr(r)rU

- H"sp p"I'Il uo J o1 lcadse.r q?l^{ sturoJ crqdrotuolne ctqd,rotuoloq 3o

(t'tt)eV eceds eq1 uo lcnpo.rd rauul u€I?IturaH lecluousc € augep e,lr '11e 3o lsrtg

BImuroJ Euranpoldall puu +cnpord TBIBcS rrossralad '1-'1-'L

uorlra[ord u€rutJag PUB ]rnPord TBIBJS uossJalad 'T'2

t8Ilf,nPord r"lef,s uossrelad 't'^L

184 7. Weil-Petersson Metric

We set

Ka(z,C) = i r p^(C), z,( e A. (7.2)n=0

Then Ka(2,.) belongs to A2(A) for any z € A, and satisfies the rvproducing

formulap (z ) = (p ,Ka (2 , . ) l a , z C A ,

for any 9 e A2(A); that is, /(a is the reprodacing kemel for A2(A). Flom (7.f )and (7.2), we obtain

@ o

K 7 1 ( z , C ) = i ? @ * 1 ) ( n + 2 ) ( n + 3 ) ( z d = - 1 2 , , , . z , ( € A .- - A \ - r ! / ,

^ U _ U " r ' "

| ^ , / \ ' r I 3 , f \ ' " t v , r \ ' s , - n ( l _ Z 4 ! ) 4 ,

Simila,rly, we consider the Hilbert space A2(H) of holomorphic functions ry'on .EI such that

ll,tll, = Ilr^rr'r-,1,t

(,)1, dxd,y 1 x.

The M<jbius transformation ?: H - A given by T(z) = (z - i)/(z f i) inducesan isomorphism ?* : A2(A) - A2(H) defined by

T'(p) = (9oT). (T')2, I e Az(A).

Hence the reproducing leemel Ks for Az@) is given by

Kn(z,e) - Ka(r(4,r|))T(ffr'111' = y+. z.( e H. (7.3)7 r ( z - C ) n '

r ' 5

Note that I{6 and, Ks are inva^riant under.4ut(4) and Aut(H), respectively.For example, Kg satisfies

Kn(" ,Q= Kn( t (z ) ,1$ \1(St ' (e )2 , z , (eH, (7 .4 )

f o r a l l T e A u t ( H ) .

Theorem 7.1. (Reproducing formula) Eaery 9 e A2(H,f) satisfies

p(z)= [[ xrfCl-',p()rciQda€a,t, z€H. (z.b)J J H

Proof. It is sufficient to prove that

,1,(r)= [ [ ^rcfr l 'G)@aea,t, zea, (7.6)J J A

for every rb e Az(A,l-'), where l' = TIT- 1. Since R = H /t is compact, ̂-'lrltlis bounded on 4. Thus the integral in (7.6) converges absolutely for all z. Bythe mean value theorem for a holomorphic function, we have

-q3rau e

(o'r)

(8'z)

rJ)L'V)z 'r(z),1((')L)i

3 =k){O

les am 'V uo { uor}3unJ crqdlotuoloq e ro; 'leraua3 u1

.v ) 2,"(z),1((z)L)/'? = ?)fi

se ue??rr^r sr fl pu" 'v Jo v ernsol? aql Jo poorlJoq

uo crqd.rotuoloq q ./ uorlcunJ eq1'V ur lceduroc ,t1a,rr1e1ar sr dr e)urs

.bpJp(),2)v:r())d._o)v "fl = evlas ellr '1t,r'oN

I Jr,t'Jj! . "(z),1 lorw QJ,Wz())4,_o)v " I | ) 3 =

l- 4771 rJ)L

"(z),(,-r1lt'p1pQ'�G),-r),>t o)f ,-o)v "|

| ) 3 =

t ptp zl()),Ll (A!:yZ(()),t)@ z - (( >l rlr"' I I'?

=( r\Ln n rJ)L

ttplp O,z)vx e),fur_e)u "

I I Z = e),t,

.0 - z roJ splorl (g'Z) elnuroJ leqt stress" slq;,

't plp()'ilvx Q)fr "-Q)u" [[ =JJ

tnlee)fi,("|)l-r)"il f = tolo

s€ uellrr^rar sl (g'Z) elnuroJ ueql'y ur lceduoc d1arr,r1e1a.r sl d leql os y ur tJ rol3{ ureruop Ie}uauepunJ e e{ptr

sarras gJBcurod z.I'1,

O 'V ) z II€ roJ sploq (g't) 1eq1 s^\oqs srqtr

' "(o) ,r ( apnp (n ' z)u ;t @)fi "-(qr" [ [\ =

\ - -

JJ /vff

apnp "l(n),sl ((^)g' (,)il, >r (@)il'h "_((.)g)v I J = @",fu

uletqo a,rn ', _,0 - 9 3ur11nd pua ofi oI (L' L) Surflddy ' "6),L(r)fr = (g)"fr

seqsll€s pue'(L,,1 ,_L'V)ZV Jo luetuale ue sr o4l ueql '.(,.!) .(L",lr) - ofr pse$ lI'z = (O)t qll/{ (y)tny 3 L luauale u€ esooqr'V ) z fue ro3'aao11

Q't)

98IlrnPord r"Ff,s uossratad'I'2

186 7. Weil-Petersson Metric

We call this Of the Poincar|, series of / for l-/ of weight -4. Similarly, we definethe Poincar6 series of a holomorphic function on I/ for a Fuchsian group l- actingon H.

Theorem 7.2. Let f be an integrable holomorphic function on A, i.e.,

[ [ v a y d n d y < x .J J A

Let Of be Lhe Poincar| series of f for a Fuchsian group l ' acting on A. ThenOf conuerges absolutely and uniformly on compact sels in A, and belongs toA2(A, r').

Proof. Denoteby B(z,r) the closed disk with center at z and radius r. Takeany compact subset K in A. Since l-' acts properly discontinuously on .4, andsince every element 1 e f' - {i.d} ha.s no fixed points in 4, we can choose asufficiently small radius r so that l(B(r,r))nB(z,r) = 4 for any z € K and

1 e r ' - { i d } .On the other hand, lf OQDy'(r)2 | is a subharmonic function on 4, and hence

the mean value theorem for a subharmonic function yields that

lr oQ))t' QY l = # | l "u,u lr (t(cDt' G)' l a'ea,t

r f f= # J J.,rur,urlr K)l a€an'

Thus we get

D,rr (te))r' e)' r s # F*, 1 1., *r,,,, r r G)t dtdn

s1 [ [ v rcyd , (d ,q<a, z€K.= r r 2 J J a '

Hence, @/ converges absolutely and uniformly on K, which in turn implies that

/ is holomorphic on 4.Next, for any 6 € l-' we have

o f (6(z))6, (z), = D f 0"6(r))t, (aQ))2 A' Q)21e l '

= I fe"6(r00"6),(r),1 e P

= Of (z) .

Therefore, @/ belongs to A2(A,f'). tr

From the observation preceding Theorem 7.2, we have the following corol-

laries.

(rlz)

(orz)

?es '(J'n)Jl ) I fi.r,olpq.ro uD rol g'Z uraroaqtr,

'Q'U) Jl ot Pepuetxa s\ (,t'tt)zv uo u( ' ' .

)lcnpo.rd relef,s uossreled arlt 'reqlrnJ 'Q'H) Jl to acedsqns pesolc e sl (J 'H)zv'lcedtuoc sr Ar ecurs'rrrrou $ql q1ral aceds qr€usg e sauroceq (.f 'g)iZ ,r"ql

' * > l(4 !lz-Q) Hudfr?;" = -ll/ll

qtl,llt I/ uo J ol lcedserr{}l^{ 7- lqSraaa Jo sruroJ erqdrourolne elqsrns€eu II€ Jo }es eq} eq (l'U) &l l"l

.1 3 L,H ) z,(r)t =,e),t((z)t)tsausIlss

pue 'Il uo uorlcunJ elqsrnwaru s sl 1l JI /i uo ..1, o1 lcedsar qlpa p- lq3rer'r gou.r,ol cttltl.totuolnv elqornsoaur, e pell"r sr H uo I uorlcunJ penler'-xaldruoc y

.(n)zV to! Hy leure{Surcnporda.r eq1 Sursn fq g uo slerlueresrp crlerpenb elqsrnseeu luuuoq 1f p - geceJrns uueruerlf eql uo slerlueraJrp crlerpenb crqdrouroloq lrnrlsuor ileqs eA\

uorlcafor4 ueuErag g'1'2

'(g'g ue.roaq;;o;oord eql 'Jc) 11 Jo sles lceduoc uo .{lurro;run

pue dlalnlosee zz/l o1 se3rerruoc J rol I lo lO serras are)urod eql uaqJ,'((y- t)i - r)t)/6 - V) = Q)l fq uaLrS .;1, uo uortrunJ crqdrouroloq alqerSelur ueaq / le1 'H uo 3ur1ce dnorS uersqcnd € sr qrlq/'\ 'I < y qll,lr zy - (z)oL f.qpelereua3 dnor3 e aq J laT

'sarres erecurod e ;o aldurexe ue errr3 e11y'a1du-tnxg

'H u? J .tot utoutop loTuauopunt o s? ,I ataym '11 ut sles Tcodu.toauo filtu.totrun puo fr.yaTqosqo safileaun eprs puoy 7t16t"t, e1l uo saxr?s eql puo

L H > z' "(z),1 luo*, Q' e)q r rr-Q)a z-e) H u " ill 3 = Q)d

tu.tol aq7 u, u?llu,rr" s! (J'H)zV 3 dt fi^raag '6 riregoro3

'V uo t@ - o\ tottt V?ns V to pootltoqrl|tau o uo peu{ap

{ uorTcun! ctyd.r,oruoloy D c?swe NeUl'(J'V)zV ) dt fr^taaa ro,tr 'T fte11o.ro3

Jf ftrplp ()'z)nx())/._())sv

J J = f,lu

eJeqm

'H uo 6O - {zd

sa{st7os puv '(J'H)zV o7 sfuopq lzd uaqJ

Hf f'H > z'ttplpO'z)n>tO)/r_O)ffv I | =?)Urg) -

JJ

r.8 Ilf,nPord r"l"f,s uossralad 'I'1,

188 7. Weil-Petersson Metric

and F is a fundamental domain for I in H.

Proof. From ll/ll- < oo and forrnula (7.3), we see that integral (7.10) convergesabsolutely. It is clear that fzf is holomorphic on 11. From formula (7.4), weconclude that B2f belongs to A2(H,f).By an argument simila^r to that in theproof of the Corollary to Theorem7.2, we get formula (7.11). tr

It is easy to see that B2: Lf (H , f) - Az(H, f ) is a bounded linear operator.The reproducing formula in Theorem 7.2 implies that B2 is the identity mappingon A2(H,f). We call B2 the Bergman projection of Lf (H,f) fo A2(H,f).

Rernark. We can also use the .Le-norm (C ] t) instead of the ,L--norm. Namely,let L\(H,f) be the Banach space consisting of all measurable automorphic forms

/ with respect to I on 11 such that

l lfll i= Ilr^'r,r'-'olf (,)lo dxdv 1oo-

Then we obtain the Bergman projection from Llr(H, f) to A2(H, ,i-) (equippedwith ,Ls-norm). For details, we refer to Kra [A-58], Chapter 3, $$2 and 3.

The Bergman projection is a self-adjoint operator; that is, we have the fol-lowing assertion.

Theorem 7.4. Ang two elemenls f ,g e Lf (H,f) satisfy

(1rf ,s ln= ( f ,1zc)n.

Proof. Take a relatively compact fundamental domain F. for f in fI. By usingformula (7.11), the transformation relations for )s, Kn, f , and g, and Fubini'stheorem, we get

(02f, cl nr t f - t r I

= [ [ ^ \ n ( , ) - 2 | D t f ^ ^ H ( o - 2 f ( 0 M d $ r 1 | 7 , Q ) 2 g Q ) a x a yrrF l tTrt t r J

[ [ , r r t - 2 t r r t - " l - t r - e - - - , , l- J lr^n\s/ , ' . .) | E/,

\se)-z@l,e)z Ks(ve),edxayl aurtLtert

t r if l= J Jrls(o-'l(o

X

= (f,?zcl n .tr

E I l,\H (z)-2 s(z)ffi 4 a,av0-')'G)'d€dn

' *(J'H)zv = ((t),D1t

'ro1nc4.rod uI ' *(J

' H)zV oTao (1) 71/(l

'U)g to tustrlil"rourost uo sectuput

nV = Qt)V fi.q uaar6 *(J'H)zV * (.1'H)g ty |urdtlour ?yJ g'L tuaroaql

'Il ul J roJ uletuop leluaur€punJ " sl d pue J/H - g"a.ra11

'(J(H)z,V ) d't 'npap(z)dt(z)d"[[ =a(6'rt) = (dt)dyTJ

fq paugap (l'U)zV uo leuollf,unJ r€eull e''a'l '*(J '

H)zV ) /y luaurale us qll,rr (.t ' U)A ) r/ fre,re elsr?ossy '(t '

tt)zv p

,(J'H)"V eceds lenp eqt qll,rr ((.t),D',1fllecruouea fg11uap11eqs a,r,r, '1xa11

(eu)' Q) n I Q' H)s = (Q).r)'.r,

el"q a rn 'oOray = (.fhf Eurllas 'snql'((.i')ag)'Jol H

uo J roJ sl"rtueraJrp rurerlleg p ('J'H)g aceds aq1 ;o Surddew .reaurl a,rrlcelrnse sr lurod as€q aql ?" O uorlrafo.rd ,srag ;o

04; airrle,urap eql 'g raldeq3 ;o6$ ur uaes s€it{ sv'1urod aseq eqt 1e (.t)alJo'(J'*H)zV = ((;)ag)og aceds

lue3uel crqd.rouroloq aW se ((l)D",t p.rc3a.r a,r,r '(.7)ag qll/" (J)J Sutfgluapl'1urod aseq eql le (,f),2 fo aceds lua8uel ctqd.rouroloq at{} ((J)J)',2 fq elouaq

lulod aseg aqt 1e aced5 lueEue; oqJ, 'l'Z'L

'aceds

rellnruq?ral e ;o saceds lua3uel eql roJ uo.Ileluasardar lrcqdxa ue a,rtE 11"qs o1t

sarsds rellnl'IlqclaJ, Jo f.roaq;, lBtutselruVuI'Z' L

(zr t)'H ) z'(z)l4"oz- = Q)(dHru)zd

uplqo em 'uorlreford ueurS.rag eql Jo uolt$Sep aql fg '("f ' H)zV Io lueruele wr sr

t"plp19,...............�, 1) .[[ "-- = (z)l4o --rJr

O)rl tt 9 -

l€rll eas e^1, snqJ, '(J'*H)"v o1 sSuolaq qarq.n

' *H ) z't ptuW"[f ;- = ?)[4'Q

eqr ur o uorlce rord ( sras ro 1,r1,a ",'11'f$":l?t1trtff.'"#511 ;;,lt Tjffi:T68rsaredg ra11nruqrral;o froaqa 1eurrsatruyq

'Z',

190 7. Weil-Petersson Metric

For the proof of this theorem, we need the following lemma due toTeichmiiller.

Lemma 7.6. (Teichmiiller) An element p e B(H, f) belongs to N(f) if andonly if A, = 0, i.e., (tt,p)n = 0 for all 9 € Az(H, f).

Proof of Theorem 7.5. Cleaily, z1 is linear. Teichmiiller's lemma asserts thatKerA = N(l-).Every / € A2(H,f)* is written as / = (.,rlt)n for some ry' €Az(H, f ) , where ( . , . )n denotes the Petersson scalar product on A2(H, f ) .Putting p = ^;rb,we see that p € B(H,f) and Au = f .This shows that z1 issurjective. Hence, by the homomorphism theorem we conclude that ,4 inducesan isomorphism of B@ , f ) /N (f ) onto Az(H , f)* . D

Proof of Lemma 7.6. We consider this lemma in the unit disk 4 instead of fI.Take a Mobius transformation ^9 given by S(z) = -i(z+l)lQ - 1), which^sends4 a"nd A* to H and f1*, respectively. Here, 4* is the exterior of 4 in C. Weset l- l = S-rfS, a Fuchsian group acting on d and 4*. Then p e B(H,f)corresponds to u € B(A, ft) defined by

u ( z ) = p ( S @ ) @ - , z € A .' " 5 ' � \ z )

Further, O"lpl e Az(H*,1-) corresponds to iL e A2(A*,f') given by

irr(z) =tb,1p1151'115,1272, z e A*.

Now, formula (6.8) in Chapter 6 is rewritten in the form

v(z) = -* ll"ffiauao, z e A*.Thus f is expanded in the form

where

1 €

i l r ( z ) = - 1 t d n z - ( n + 1 ) , z e A * ,' 7f u

n=3

f lan = n(n - lX" -r)

JJ^v(()("- 'd(dn.Hence, f = 0 if and only if an - 0 for all integers n 2 3, which is equivalent

to the condition

[ [ ,tclrn dutt = aJ J A

for all holomorphic functions / in a neighborhood of 4 . By the same computa-tion as that in (7.8), we obtain

[ [ ,rclro dtd,t = @,of)n.J J A

Therefore, by Corollary 1 to Theorem 7.2, the lemma is proved. n

Gz'D '(t'u)a = ,t '14"o = l[,t]ul'ourelqo a,ta (qI'z) ruo.r; '.re1ncr1red u1

6z'D '(t'n)a )'t '[4o\ = tt4n)a

1eq1 fldurr elnwroJ Surcnporda.r eql pue '(tt't)'(gt'Z) 'raaoarotr41

(O;r) '1t'11)zy)d 'ldl,l -ffa]An

1aB an 'elnur.roy Surcnporda.r aqt pue (gt'Z) '(lt'Z) ,(q 'reqlrng

'(,f),rf - F/raX

?sq1 apnlruoc a,rn '(21'2) pue (91'2) uorg

(er.r) .(,t'u)a ) ,t ,(dHru)zd rlU = l\n

ot peel (tt't) p"" (91'2) selnutrog

'(t'n)gu - (J'H)s : H

Surddeur reeurl elrlrelrns e urelqo an '.{€.tr srql uI'[r/]g qerluareJrp rurerlleg f,ruorureq asaql II€;o aceds ro?)al eql (J'n)g U fqelouaq'r/ fq pacnpurloquataStp tu,rorneg zruontrvq ar{t eq o} pl€s u [r/]g stqa

et t) .(z)lr4dt"_(z)ry = ?)l,tlH

,(q (,f 'A)A > [4n lueurela ue eusep aa\ '�(J 'H)g > r/ drare .ro; 'txaN

'11 t p)zy ) d'6 = [[6]illdl

eABq air 'e1mu.ro; Surcnpo.rde.r aq1pue (7I'l) urorg'(.7'H)eV ol(l'n)Jl3o uorlcefo.rd ueu3rag eql sr z5l a.reqm

(gt.r) ,(t,n)g > ,t ,(dg"U)zg = l4d

sp1ar,( (61'2) elnurroJ 'g'I$ Jo {reureg aql ur pal"ts s?.r\ sy

(qt'z) 'H)z '(z)lrt)"?7-=Q)ltt)d

fq uanrS (L 'n)zV

) 146 lua{uale u€ q}-r^r (l ' n)S 3 r/ {.reaa aler)osse arrl 'pueq raqlo eq} uO'd .{q pacnpu p4ua.taStp truDr?I?g ?tuoulrvU eq1 l6ld $q} IlBc a14

&;D 'H)z p)c!"-(z)Hy=(z)ldlrl

fq(l ' U)A ) ld',lrt eugap a^r '(J ' H)"V ) d lueruela ,{.raae .ro3 'arag '(.7)g

3o srotcallue3uel luase.rde.r ol sprlualegrp rurerllag ?ruorur"q asn a1yalor luelrodun uepa,{e1d (l'n)ev Jo slueuale uorJ peuuep sler}ueraJrp rtuer?lag rruoureq pell€l-os 'g'Z ueroeqJ pue (6'9 uraroaq;) ureroaql illa6-sroJlqv eq1 ;o s;oord aql uI

sIBrluaraJIC rurerllag cruourrlall'z'z' L

16I sacedg raflnurq]ral 1o i(roaq; Furrselruyul 'Z'l

192

Using (7.17) and (7.20), we also have

H 2 = H . (7.22)

With these prepa,rations, we get the following assertion.

Theorem 7.7 The space B(H,f) of Beltrami differentials for I on H is thedi rcc l sum of HB(H,f ) and N(f ) , i .e . ,

B(H, f) - H B(H,f) o N(i-) . (7.23)

The deriuatiue iDo of Bers' projectioniD al lhe base point iniluces lhe isomor-phism tb o : H B(H, l) - T,(TB(-I-)). /n particular,

r"QggD= HB(H,r) . (7.24)

Morcouer.

j t , p )n= (H lp l , p )n , p e B (H , f ) , p e A2 (H , f ) . ( 7 .25 )

Proof. Take an element p € N(i-). Since N(l-) - Kerf/, we have H[p] = 0. tfp e H B(H,l-), then by the definition there exists an element v € B(H,f) withH[v]= p. Hence, (7.22) leads to

p = H [ v ] - H 2 [ " ] - H [ p ) - 0 ,

and we obtainH B ( H , l - ) n N ( l - ) = { 0 } .

Every p e B(H,f ) is decomposed into

p = Hlpl+ U' - HU'l),

and (7.22) implies that pr - IIljtl € KerIl = N(l-). Thus we have (7.23). Itis obvious from (7.f3) that @o: HB(H,I-) * T"(?:BQD is an isomorphism.Accordingly, we have (7.24). Further, Teichmiiller's lemma (Lemma 7.6) yields(7.25).

7.2.3. Tangent Space of "(.f) at a General Point

We shall give a representation of the holomorphic tangent space Q(T(l-)) ofT(f) at an arbitrary point p = l-'1.

Let T(f') be the Teichmiiller space of l' = w' l(w')- I . Then the tra.nsla-tion mappinglw'l* ($2.3 of Chapter 6) of "(.l-) toT(f') induces a,n isomorphismof Z)(f(f)) b T"(T(f')). We give an explicit desciption of this isomorphism.

Defining / \

rc = F(.\) = [ % ! -!.

] o(,,)-',\ ( t a ' ) " r - ' ^ l .

we have

7. Weil-Petersson Metric

D 'uorlJess? puoces aql a €q

a,ra 'd;o rol e^rleluasa.rda.r e Jo e?roqc eql Jo luapuadeput ere li Pu€

r4i acutS

'H ) z '(z)(l4"go 4)"-(z)u14- =

(z)(14 ^t " "( ̂9)) z-Q) u ue- = Q)lttl(^1 o ̂11)

1aB aar '(tt't) pu" (gl'f) urog 'raqtrng'ursrqd.rouosl rre sl (^J'tt)gtt <- ̂6rcyf (J'H)g i n'I o nH

1eq1 seqdurr tualoeql ursrqdrourouroq aql 'acua11 'anrlralrns sl nI o zll ]"ql pue

'ngrr>[,= (^go 4i)ray = (nI o "(^O))raX = (n7 o ng)rcy

leql aes a.l!r'crqd.rourosr are 4 pu" a,I Wql pue'o(r4i).reX -

,IIraX 1eq1 3ug1ou'snq;,

^ '((^.t)a,t)",t ((t)sJ)oJ,

"(^o)T I ", 'tl

GJ'H)B .- (t'n)a^1

:ure.r3erp elrlelnruruoc aql a sq am 'sarttleltlep Suqe;

-^ Q)s,t,

1" I

{- r(l'n)g,I

:urerEerp all?slnturuoc 3ut,ra,o11o; aqluplqo e^t ueqJ'flalrlcadse.r'r(nJ'H)g pue r(J'H)g Io suorlcalord (srag eq

ni1 pue A p.I'(^J)aa + (tr)aa i*\nnl Surddeur uorlelsuerl erll 4i ,(q elouaq',(1a,rr1cadser'(^J),l,pue (J)JJo peelsul (^J)s1 pue (;)aS reptsuo? e14'{oo.r4

'd Tutotl ayq to nm ea4oTuasa.tdat o to enoqc ay7 to

Tuapuailapu sl nI o ng 'taaoano141 'rt 7o uotTcato.td,s.tag lo eatToauep arfi s? ng

ataqn '((n1)J)'.t = (t'n)g H ot ((l),f)d,t = nOny/(,t'n)g lo utsttld.totuostuD s, aI o ng |utddout eW'(J)J > l^^l

- d'fitaaa rotr 'g'L uollrsodor6

'uollress€ Sur.troloyaql e^eq e1t{ ueqJ,'Qt'D fq ua,r6 $ qstq^r'Il. uo nJ to! qelluereslP rurerllegcruoureq Io (nJ'n)g U eceds aq1 ol (,J'H)g Io uorlaalo.rd elq1 nH fq alouaq

'(^J)aJ

"'lr(nJ'H)fl

@z't)r-(.'). (+ #) = 6r -&?rdiwl = t4 ̂ti(q ua,rr3 usrqdrouost ue aq (r.i,'H)g * (l'n)A : nI Ia'I

'(.r)z > [tm'1 '[,t] = ([tr-r])'[,rn1

t6rsacedg rallBurqf,ral ;o froaq; Frursalruyul'Z',

194 7. Weil-Petersson Metric

Remark.In the rest of this chapter, we identify fplgD and ""("(f ')) with

B(H,f)lKer6, and HB(H,f'), respectively. We also identify Q("(f)) withHB(H,f ' ) under the isomorphism H'o L ' .

7.2.4. Connection with the Kodaira-Spencer Deforrnation Theory

The subject of this subsection is not needed for further development in thischapter. However, it is interesting by itself. We shall deal with the tangent spaceof T(,l-) from the viewpoint of cohomology theory.

We recall the fundamental idea of Kodaira and Spencer on the deformationof complex structures. For details, we refer to Kodaira [A-57], and Morrow andKodaira [A-77].

As was stated in $1.1 of Chapter 1, a Riemann surface r? is obtained bypatching domains D1 = z1(U1) in the complex plane. The identification betweenDi and D1 is given by a biholomorphic mapping zjk of an open set D* j = zp(U1n

U*) C Dp onto Dip = z1(UinU*) C Di. A deformation Rr of r? is consideredto be the gluing of the same domains D1 viaa different identif ication /ir(.,1),where /i1(.,1) is a biholomorphic mapping of Dpi onto Di* with parameter I =

(1, , . . . , t - ) such that f ip(2p,O) = z1p(zy) . I f a l l f ix (zp, t ) are C- funct ions, weget a differentiable family { /?, }, of Riemann surfaces. In particular,i l f 1yQp,t)are holomorphic, we have a holomorphic family of Riemann surfaces. From hereon, we consider a differentiable or holomorphic family of closed Riemann surfaces.

In order to know the actual dependence of the complex structure of.R1 on theparameter t, we consider its infinitesimal deformation as follows. For simplicity,we assume that m = 1, that {Ui}i is a locally f inite open covering of R, andthat every Di is an open disk in the complex plane. Take the differentiation of

fi*(rp,t) with respect to I at t = 0. This is regarded as a holomorphic vectorfield on Ui f\ Up, which is written as

, } f ; r , ^ , 4t1* = f i ; (z r 'o ) a r i '

zk = zk i (z j ) '

The relation fir(fxiQi,t),r) - zi on Ui fl [[ gives

? i n * 0 6 1 = 0 o n U 1 f l U p .

Further, the relation fii ltt,t) = f 1t(f *t(tt,t),t) on Ui fiU* i [/z yields

0 i * * 0 u l 0 q = 0 o n U 1 f i U P n U 2 '

Thus it follows that d = {01x } defines an element [d] of the first cohomologygroup //1(R, @) with coefficients in @, the sheaf of germs of holomorphic vectorfields on .R. For the cohomology theory, we refer to Gunning [A-40].

This [d] represents in some sense the derivative of the complex structure of/?r with respect to I at I - 0, and is called an infinilesimal deformation. of R.We call Hr(R,@) the space of infinitesimal deformations of ,R.

Eur11n4 '!17 uo pleg rot?e^ *C s sl qcrqn '(fz)fr,z/((lz)lrzSlto((!z)nz)rd tf- la ''e'ltlqeqd q3= lre/gfo las a/vl ''tt)u ln uo pleg rolcel crqd.rouroloq e srlzg/g(fz)tlp - r!6 qtea ?sql qcns {'t!0I = B alcfaoc e dq paluase.rde.r sr qcrqirl(O'A)fl > [g] lueurala {ue a1e;'UJo { lp} Surreroc uedo eql o} el€urproqnsflrun yo uorlrlred e eq { !6 } 1a1

'sr'ro11og se ua,rr8 sl *9 Jo Surdderu asralur eqJ'{o'a)tn ot

((t-r)0,03'a)ogQl(G-")r,03'a)oH;o ursrqd.rourouoq e se,rr3 [p] = ([4)-g dqpauuep *9 Surddeur aqr r€ql realc sr 1I '((r-v)o,og'u)oug/((,-v)r,03'u)o1l ulrl p lrll ssel) acuel€Arnba eql uo fluo spuadap [A] qq.f, '(O!)fl yo [6] luauralau€ se)npur {'t!0} = 0 ptrr-'rtng !2 uo plag rolce^ crqd.roruoloq .s fl rfB uaqJ,

Qz't)

les aM 't2U ln uo seqslu€A lzg/gla - rzgf gtafra,re 1eq1 fl U uo pleg rolra^ e sa,rr3 {!zg/gla} teUt uorlrpuo? }uar?lgns puef.ressacau V'U uo pleg rol?a^ *g 1eqo13 e eugap sfernle lou seop {ftg/glo}'rarreruoll '12 uo pleg rolla^ -C € sl lrQlglo ueqa 'f4 uo lrl = lggf laguotlenba Flluareglp eqt Jo fo uorlnlos e a{€l '{fzpl[Zplrt] = il.{ue rog

'!ag/lag = frl qur* {lzp/lzpfd },(q paugap sr og pue'g'uo {lzglgfa}

- o plng rotrel -C e f,g uarrr3 sr ((r_")0,03'A)oH 3 a luauraleuy'lI uo l€Ilueraglp rurerlleg -C e ''e'l'U ro (I'1-) ed,{1 p {!zp/fapfrt}- r/ uno; IerluereJrp -C € sl ((,_r)r,03'A)oH Jo luetuele uB ?eq} eloN

('tualoaql s(1ln€eqloq;o ecuenbasuo? € sI srql) 'fe,r 3urmo11oy aqt ul

, (('-v)n,03'A)oHg . ^ (o'a),u -# ' (\ '-' ) ''n3W)oH ' *t

usrqdrotuosl eql lcnrlsuol eA\

'((")o,rO'A)oH = (A)zV pue '(r_")O - O

a^?rl allr ueql

'!, Jo suollces-ssolr er€ q)rq1rr senl"A qllllr

surroJ-I ctqdrouroloq;o sur.ra8 Jo Faqs aW - (y)o,rC)

pue 'r_v Jo suorlres-ssorr er€ qcrq^\ senl€^ qlr^r

(1 'g) ed,t1 Jo surroJ lerluaregrp *C Jo sur.raS;o Jeeqs eqt = (r_r)r,03'r_y

Jo suorlf,es-ssorc *C Jo sura3 go Jsaqs aql = (r_r)0,03'r_v

Jo suorlres-ssorc crqd.rouroloq;o sur.raE Jo Jeaqs aql - (r_!r)O

:uorl€1ou aq1 ldope e1t '{n g lp uo 1l,r/t fq ue.tr3 srrfv uor?cunJ uorlrsu"rl sl-r ''e'l 'g, uo alpunq eurl Iecruou€r eql aq x p1 'too.t4

'yutod

esog eqt p (A)J acoils u11nu,ycpal aqT lo ((U),D"1eocds Tua|uot ayl qpn p?*l-uep, st A uo suorlout.totap Tou,nsal.?u{u?lo (O'A)rn acods aqa.6.2 urarooq.1,

. lzo - - ttzo.'qnU ln uo ---:-(!z)!a - :" (cz)'ta - 'tte

ea

96Isacedg rafl]urqf,ral;o froaq; Furrsalusul 'e'l

196 7. Weil-Petersson Metric

pi = AailAzi, we get an element p = {pidzildri} g Ho(R,t0'r(rc-1)). thenthe homomorphismof ^I{1(.R, O)to Ho(R,to,r(n-t))/6Ho(R,8o,o(,6-t)) sending[d] to [p] gives the inverse mapping of d*.

Next, we have a canonical isomorphism

where

, ,1o (6 ' ) - 1 : HL (R ,o ) * A2 (R) * ,

A : Ho (R, to,L (n- | \ I 6 Ho (R, t0,0 ( , i - 1 ) ) - A2(R).

is defined by

t lt lp)@) = I I pQ)pQ) drdy, p e Ho(R,to,t(^-t)), e € A2(R).

J J R

(This is a consequence of Serre's duality theorem. See the proof of Theorem 7.5.)Since 7}("(r?)) is isomorphic to ,42(,R)* (Theorem 7.5), it follows that the

infinitesimal deformation space H'(R,@) is identified with the tangent spacef"Q@D of the Teichmiiller space ?(.R) of .R at the base point. This completesthe proof of Theorem 7.9. D

Now, we wish to see that H'(R,@) is canonically identified with a subspace ofthe first Eichler cohomology group I11(l-, I/2), which is defined later, where f is aFuchsian model of -R, and I/2 is the space of polynomials in one complex va,riableof degree at most two. Note that IIz is regarded as the space of holomorphicvector fields on the Riemann sphere C. Further, If2 is canonically identified withthe Lie algebra sI(2,C) of SL(2,C), which is the tangent space of S.L(2,C) atthe unit element.

Let (H,r, ,R) be the universal covering of .R with covering transformationgroup l'. We use the notation:

B(H,r) = the lift of .Ho(.R,to,t(^-t)) under er,

V(H,r) = the l i f t of Ho(R,50'o1r-t)) under r .

Then 6(I/, .l-) consists of smooth Beltrami differentials it for I on I/. An elementof 6 eV(H,l-) is a C- complex-valued function on 11 such that 6o7f 7'= 0 fora n y T € f .

For every 6 eV(H,f), we set 60 = 7itf 02, which belongs to B(I/, j-). Obvi-ously, f/0 (,R, to't (*- t)) / A Ho (R,go'o 1r-

t )) is isomorphi c to B(H, f ) I AV @, D.

For any it e B(H,l-), we put

^ , \ ( F Q ) , z € H/ r ( z r = 1 0 , z € C _ H .

Let F be a continuous function on C which satisfies the differential equation

f f= , on c (7.28)

in the sense of distribution such that .F(z) = O(lrl2) as z e oo. For example,we see from Lemma 4.20 and Theorem 4.37 that the function

elaqAr

, lr_,il=' !_1,s= = (c r, J ) | H (zII'J){1es aair 'mo1q

'QZ'D,(q paugep B o1 spuodsallel [/)/X leql slsaE3ns slqJ

'r)L 'i=Wlf -h=@)t!ltY

ultslqo a/rr snqJ

',IIil{r(=Lo[rfll

aleq a^r'J > L f.rara rog rtotl-- l,o!l"acurs'1/ rog 1er1ue1od e sr [r/l/'sI

lpql '(AZ'D,tq uaar3 sl 0 = l l" I ol lcadse.r qlp y lo ltlll uorprluaragrp a{1'7 raldeqg q /t'7 ueroaqJ ruo.1{ '0 --

I Ie I o1 lcadser qll,t{ ,,! Jo e^rle^lrep aqlL dq alouap 'J ) L frara rog '{(t'.yf1} o1 spuodsa.rroc r.7 qql 'g .re1deq3

Jo I'I$ ur pal?nr?suoc sr rlf,rqtt'r/1 luarcgaor r.ueJllag q?l^r C 1o dtm SurdderuprrrroJuo)rsenb aq1 sr t/ 'a.ra11 '{.1 > L ,

,JIoLor! - tL | *I = ,; pue (ff )T

- rg17 qfli* {r,t/rU } ,(t1urnl elq"rluereJrp e eleq a^'(J'H)g > r/ f.rarra rog'a - (a).L = (t)(a)g lsrtl q)ns zil orul J

yo Surddeur eq? .{q uanrS sr q)lqr'r d p fi.lopunoqoc eq1 q (a)g araqar '(4.)9-p.rX- cX sn{I'til ol sEuolaq I - C - d ueql'r/ ro;1er1ua1od reqloue q rJI

'22� )d'J)L '#;=61-r

e\^ zil uo slce .7 'a.ra11

.J ) zL,rL ,(zt).tx + ((rr)rx).(zt) = (zLotL),tX

uorppu@ a1cfrcoc aql sagfll"s rlorq^\

,zII * ,1 :.{XSurddeur e urelqo

ain 'acua11 '211 3 (L).rX l"ql s^tolloJ l! '/ roy 1er1ue1od e oqe sr ,1"/l"og ecurgI

'J)L ',t- ii=-W)rx ,vo,{

las art 'lroN'J ) L {ue.rog saqsru€A f, -,Lf Log leqt sr (l'n)A, f d }eql uor?rpuof, luerclgnspue f.ressacau V'(J'H)ft ul peur€luor sfe,ra.p 1ou sr / roJ dr lerluatod y

reqr s^rorror r! ,oo * z se ("lzns:ili;"r 6;lf,",Xj:Jt#Tj,t'"H'i,f.;fl -oloq e sl .{ - O l€rl} si$oqs Btur.ual s,1fa11 'rl to1 C prlualod reqtou€ rod'r! ro11o4ua7od ue g u€ r{ens ner eM '[gg-V]

€rX uI 41 .ra1deq3 Jo ''I "tuurerl osle aas '(86'l) Jo uorlnlos perrsap e sarrr3

("-)Xr-)))\62'L)

LOL

0t"[l

GF-=?)tttp?p

saredg rafl]uqf,raJ;o froaq; Furrsalrusul 'e'l

198 7. Weil-Petersson Metric

BL (r, II2) -- 6(IIz),

2 1 ( f , I I 2 ) = { x I y : | + I I z , x ( t n z ) = ( t z ) . ( x ( t r ) ) + x ( z z ) , 7 t , 7 2 € l } .

This Ilr(f, nz) b called the first Eichler cohomology group of l.

Theorem 7.LO. Let B* be the mapping

B * : B ( H , D / 6 v @ , f ) - H t ( f , I I z )

defined, bg0.(li'D = [xr],

where F is a potential for p. Then B* is an injectiae homomorphtsm.

Proof. First, we show that B* is well-defined. If p,i e B(H,f) are equivalent,

then there is an element 6 e V(H,f) with t, = ir + AilAz. Let F and G bepotentials for p and /, respectively. We put

Since 0o7 - 61t for every 7 € .i-, it is seen that for every ( € R, d(z) - Qas z + ( through I/, and that d(z) = O(lzl2) as z + oo through 11. Thus Gois a continuous function on C, and satisfies AG"lAz - /. Hence, Go is also a

potential for l', which implies that X6 = XGo + 6(P) for some P e IIz. Noting

that X6. -

XF, we have 16 =Xr l6(P) , and hence [x6] = [Xr ' ] in H|( f , I I2) .

This implies that B* is well-defined. It is clear that B* is a homomorphism.

Next, we verify that B* is injective. Assume that B.([/]) = [Xr] = 0. Then,

there exists an element P e IIz such that XF = 6(P). Putting 6 = F -P' we see

that 6 is a potential for 1.r, and X6 = 9. Thus d belongs toV(H,f), and 06 = p.

This shows that p] = 0 in B(H,f)I)V(H,f). Therefore, B* is injective. tr

We shall construct a canonical homomorphism

Pt H t ( r , I I z ) - B@, r ) lAv@, r ) .

(It is considered that B* and B correspond to 6* and (6-)-r in the proof of

Theorem 7.9, respectively.)Choose a smooth function p on H which satisfies the following three condi-

tions:

( i ) 0 5 p S r .(ii) For each z € 11, there is a neighborhood U of z and a finite subset J of f

such that p = 0 on 7(U) for every 7 € f - J.( i i i ) D?€r p" t / ) - I on H.

Such a p is called a parlition of unily for l- on f/. For a proof of its existence,

see Lemma 3.1 of Chapter V in Kra [A-58].For any [X] e //t(l-,I12), we set

( F ( r ) * i l ( z ) , z e H ,G " ( z ) - t o i , i , z e c - H .

'Q 'u)a ) zd 'rrl '* lltrtos'[zrl]d,) = (lerllH'ltrt)n)rt

p1ar,{ (19'2) pue(21'2) selnurrod 'I{ ul J roJ ureuop l"lueurepunJ lceduroc ,{1arrr1e1e.l € sr dr areq^l

'fipxp (4ert (z)rd r(z)uytfl = (ed,vt)tt (re'z)

,{q paugap sl (J'77)g ) ztt'rrl s?ueuele o.tlJo lcnpord reuur eqf .(l,U)A

uo lcnpord Jauur u€rlnu.ra11 e SurarS r(q lrels er'\ acueq pue,(L.L uaroaql)(t'n)AU qrp ((J)J)o,6 pegrluapr e14.1urod aseq aqt w (7)Jl" (Q)t)"taceds lua3uel crqd.rotuoloq eql uo lrnpo.rd rauur uer?nura11 e a,u3 a.tr ,11e

Jo lsrrd

crrlatr l uossralad-Ila \ aql Jo uorlrugao .I.g.z

'rrrlal'u

ralqgy e sr 1r leql aoqs pu€ (,1)Z uo crr?etu uossraledlra1\ aql augep II€qs e A

rrrlatrN uossralad-lla^. .t.z

D '(l'n)zV oluo 5/ray yo usrqd.rourosrue sr 12 leqt pagrra^ 41I',,,t = ([X])o tes eM.[X] ssep acualenrnba eq1 uo,tluospuadep pue'(J'H)zV Jo luauela ue sr,sarurl aa.rq1 /Jo uorlsrluareJrp aql (////

1eq1 parord q tI 'J ) I II€ roJ / - ,L/Lol = W)X t€ql qcns l? uo uortrunJcrqd.rouoloq € sl g - otr -

./'"nql'(l'U)A ) g auros rct Zglgg = Zgl"lgsegsrles (Og'Z) fq pelrnrlsuoc og uorlcun; aq1 (6/.ray

I [X] .tue rcg.too.t4

'J ro{ g uo su.tto{ ctyo.tponbculd.r,outolorl Io (.t'tt)ev acods aq7 qTtm pa{cTuapt st j"tay .II.Z uoltlsodo.r4

'flaarlcedse.r's a ss o1c fi 6 o7o ru o t1 ocralqcxfl pu€ sassDl, fr,6o1oruot1oc sragr pe1ec a.re 6/ lay pue *5/ ru1 Jo stuetuelg

'd n>L O *drul = (til'J)tH

pue 'alrlcel.rns sr 6/ '(91'2 ua.roeql ]r) e,rrlcafur sr *6/

leql s 1l\olloJ II'p! = *dod Wql rselc sr lJ,*5/ pue 5/;o uorlcnrlsuor eql uord

[/] or [X] spuesqrlr{,ra (J'n)A.g/Q'H)g - (21'J)fl:6/ tusrqd.rouotuoq e ure}qo arrr ,ecua11'(eII'J)rH ) [X] sselc acuale,rrnba eqt uo,{1uo spuadap (.t,n)Ag/e,tilg "l

[r{ sselc acuale,rrnba eq} t€rl} uees sr tI.(J,H)€l o1 s3uolaq zg/"}g - r/ snql

'J ) L'o,4 -,L1Loog = (L)X

?eql q)ns _iT uo uotl)unJ _C e sr od. uaql

J)L'H > z'(z)(L)x((z)L)d 3- = G)otr (oe'z)

66If,ulaw uossreladlral|t'l

2OO 7. Weil-Petersson Metric

Lemma 7.L2. For anA pr,ltz e B(H,f), the following hold:

h(Hlttrl,nlyz)) = h(Hltttl,pz) = h(h,Hlpzl), (7.32)

h(Hlpi ,u[pzD = (Hlpi ,plpzDa = (pr,plyz])n. (7.33)

Prool. First, (7.18) and (7.31) lead to

h ( H lp rl, t' r) - (^hE, *, n 1p 1) " = (\'uE, 9 r( r ?, f, t ) " .

Since the Bergman projection B2 is self-adjoint (Theorem7.4), we have

h( H lp t), p2) = (9 z(o?n p), \'nE) n = h(y r, n lyzD .

In particula,r, using (7.22), we get

h(HIptl, nlpz)) = h(h, H'lpzD = h(pr, H[pzJ),

which shows (7.32). Moreovet, from (7.17), (7.25), and (7.31), we obtain (7.33).tr

Now, a Hermitia^n inner product on [(?(l-)) is induced by h under theidentification f"gQD = H B(H, f).

Next, we define a Ilermitian inner product on the tangent space Q,("(f))of "(f) at an a,rbitra,ry point p = lw'l as follows: using the identificationfpglD = HB(H,f') (see Rema.rk in $2.3), an inner product on Q("(l-))is induced by the inner product h' on B(H, f') which is given as (7.31). Actu-ally, the inner product of elements 11' o L' lpr) and H' o L' [pz) in H B(H, f '),

which are also considered as elements in Z)(f(f)), is given by

h'(H' o L' lptf, H' o L' lpzl), Ft, ttz € B(H, f).

In this way, we have defined the Hermitian inner product on each tangent spacerpQQD of "(f).

Now, we study the dependence of the inner product with respect to p. Forthis purpose, it is sufficient to consider dependence in a neighborhood of thebase point in "(f).

Take a basis { pi}1;5" for A2(H,f), and set

u i = t r l g i l , j = 1 , . . . , 3 9 - 3 .

Then { ,il ln=1" is a basis for HB(H,l-). Put

3g -3

u ( t ) = D r t r r , t = ( t t , . . . , f t - s ) e D ,j = l

where D is an open neighborhood of the origin in C3g-e.

fq paugap il d AV crrlaru uossJeladlla l aql Io

"^o uttol uoss-ral?d1l?/14 aql ro'ru"tot-4 loTuauopunt eqa'Q).r'

uo ?uleu uoNuuouety uossr?Ied1r?/fi aql aa6 11ec e71t 'eJoJeq uarr,r3 uorleag

-lluapl eq? repun srolcel lueEuel crqdrouroloq se pepre3ar ere aprs pueq 1q3r.raql uo esoql pu€'slolcan lua3uel Isar ale aprs pu€q Ual eql uo { pue ;g'ara11

. (A' X)a ilrlsu Z = (A, X)"^6urroJ aql ul

uellrur q "^6 ueql 'tI-gI'dd '[Og-V] $rreH pus sqls[rp aas 'spelap rog 'f, f.reraro;'flaarlcadsat'd(!zgf g)p'o(!rg/d o7 d(ng/g)'o(!re/g) Surpuas tusrqdrotuosr

lBar eql ,tqua,rr3fl uollef,glluap! eql'dpunore (t-tgt"''1 = f) lnp+lr = lzsaleulprool 1eeo1 3ur4e1 :snolloJ * ((.1)'CXZ eceds lue3uel crqd.roruoloq aq1rltlru Paglluapr sr d lurod e l€ (J)J go aceds 1ue3ue1 Iser eqJ 'dlt{V culetuuossreledlle^\ aql ,tq Pecnpul (.t); "" crrlaru ueruuerueru eql aq "n0

le.I

crrtotr tr uossralad-Ila/$, aq?;o {11.ra1qg1 'Z't'L

'(t)poW dno"rf .tolnpour r?llnuqcreJ aqt lo uotTco aqqrepun luolroaut st (.1)a uo da! )uleur uossreled-INel1 eqJ 'tT'Z rrraroaql

'uoll

-rasse SurmolloJ eql a\oqs usc aaa 'crrlaru uossreled-lraAl eql Jo uorlrugep eq1 ,tg

I={'f'llPttP(l)lfq

3 ?,= zd/YtsPe-6e

se uelllud sr 1t',tgeco1 'auq fq palouap sl prrs'(J)J ro )uleur

uossr???d-Ir?/! "qI peil"c $ (.f )Zfo elpunq lua3uel er{t uo lcnpord reuur eql'Z'g$ ul I {r€ueg ur uanE $ uorlresse slq? Jo;oord y

'q uo *Cssolc lo n lfqqcoa'sacuoTstanc.tr? eaoqo e1tr?pun'gl'l rraroaql

'rueroer{l 3urmo11o; eqt e^eq e,r,l, uaql '6,1^J /H = (l)ef araq^{

(vt' D'(r)u((3)rnld'(it,r1= ([(l)ta]1,y, 11'[O!'t)g1^H)171^tt = (7)!!q

,(q uanr3 sl [(r),orj - (7)d qlraa ((.7)g)(t)ag

> 't?g/e'!W/g srolcel lua3uel lo (l)!lq lcnpord reuur aql snql 'O 3 1 f.rarr,aroJ (1r;,J'H)gH Jo s-rs€q t q el;j{[(t)!A<l^H] teql s^1,oqs 8'Z uorlrsodor4

'(I ) t'ltn)6y^I = (7)t,t

?as e,1\'(t)al, u1 lurod e$eq aql;o pooqroqq3rau uado ue oluo 6,go Eurddeurrqdrouoloqlq e q ((t)n)O = (t)"O ,(q peugap (.t)sl, *- O i o4i Eurddeur

aql leql 8'9 uraroeql ,(q aurnsse feru em '6r leurs ,(lluarcgns e Sursooq3

(qe 'r)

IO7,rulaN uossraled-Iral\'8'l

202 7. Weil-Petersson Metric

, * r (X ,Y ) = g * r ( iX ,Y ) , X ,Y e fpQQD.

Here, iX means the real tangent vector corresponding to the holomorphic tan-gent vector iX under the preceding identification. Namely, if ./ denotes the

almost complex structure on ?(l-) which corresponds to the complex multipli-

cation by i, then fX means "IX.This c.r*, is also written as

u* r (X ,Y ) = -2 Im hs rp (X ,Y ) .

Locally, u*, is represented by

3g-3

e*" = i t hj;(r) dt i natx. (7.36)j , k - L

Note that u*, is a positive (1,l)-form on ?(f).We say that hsp is a Kiihler rnelric if the Weil-Petersson form tr-" is d-

crosed' i 'e' ' du*'=

Ut'];"t,;,r=,,. '",::T

""

(7.s7)\ t r - A i l '

r " u ' !

on D, where hi1 are as in (7.32). This condition means that hysp osculates to

order two to the Euclidean metric on C3g-3 at every point p efQ), that is,

we cim find local coordinates Zr ,. . . ,z3g-3 around p for which

3g -3

dr*r '= D @1, + aix(z)) dzi dT ,i , k -1

where all partial derivatives of air of order less than two vanish at p.

Now, we have the following theorem due to Ahlfors.

Theorem 7.15. (Ahlfors) ?[e Weil-Petersson metric is Kiihlerian.

Proof.We follow Ahlfors [7].Bv a translation of the base point, it is sufficient

to prove formula (7.37) at the base point.

We put f l - w 'Q), f 'Q) - *v( t )71.v( t ) ) - t , R( t ) - Hf fv( t ) , and P(t ) =

f ' (F ) .We se t

I { (z,O= -- l : .Q - c ) 2 'We also set

r , , 2 \ U ' ) " ( ' ) ' ( / ' )e (ot\t\z'\) =

7V1oy _ J1q1y '

Since /'(z) =TO for all z € C, it is easy to see that

EI;n = K{z,a),I {1Q,O - Kr(2,C) - Kr( ( ,z)

(oe'r)'apnp (n)ltrt)r,r^, ffi"il +- = @#*'snq5 'uars 1er?a1ur eqr repun ) o1 lcadsa.r qrr^{ pat"r}uareJlp "- ".T[Tf"T:il;leql ees er,l.'t?,'V uorysodor4 ;o goo.rd eql ur leql ol r€grurs 1ueun3.re ue . g

' apnp (n)ltrtrrrr,, W' il +- =

,rM-()),n?"r#'u3ts

lerEalur aqt rapun g o1 lcadse.r qrr^{ pa}erruereJ-rp eq uec ler8elur dlt"Htt

'apnp(n)lrnJ,,r^t l# -i +((4:!-- ^)!Q)'I - ̂)l"tt

Lr r r ltJ r

apnp(n)rrt,((rn)-(,/)) f t - (')'f -@)J * ,L I I

(9);[--(^),t)(Q),! -@)J)]"tt u _=I lll r

((,),! - trJ)aq+urelqo a,rl ,/t-t tueroeql o1 dre11oro3 eql pu€ (z)rl = (z)r/ Bursl

.H ) z,) ,((r),! - ()),/)sor 'nln = @,)),>t zU

srqr ro pua a{r 1€ a {r"ureu .r,) o:r;Tii,Ti::;t5":LJlj::,'[;;?::il;(ee'z) np,p

{up4p())ta (z)qn "(2,)),>r"l[ *] [[ =frlrt,

t

uplqo ea,r, '{{q - lfi1 uro.r; snq;,

npap(z)ta {** e)qn "(),,t,x l[ *]"il

=e)!tt1

1aB e,u '(gZ't) pue flrrl1lnI = (7)l,r')'z r,o1 Q)rt'Q)rt elnlr?sqns '1er3e1ur $ql uI

Hff v, ) (r).Irr -Q)(t)qn,()'z):r ll :l ll =(r)!!a

JJ 6L) JJ

ur 1{r1 }eq} aes e/( '(gt'z) pue (0I't) urorg 'rrro51'O ) t Pu€ *FI n H > )'z 1P'ro3

f,rrlel I uossraladTalUt'l

3ut11eoa.r'ueqa'flaarlcadser

('nPxp (z)()lrt l,tplp (

uroJ eql ur uellrr^\ s! ('g'Z)

t0z

204 7. Weil-Petersson Metric

Here, this integral is defined as the Cauchy principal value.

From (7.38), (7.39), and the definit ion of K1(2,O, we have

0h ' t

#(')= + il,{ I Lryr I{,(c, 4;ke) v1 G) du,t} daav

24 l f ( t l .= - -T i? - - - - i t -T ,u ( r ) r rz r r r r - f= - # J J, u., \ J J . o fr, z)r1e, 0T6 l,J@ 7' {t) lvl(C) d4 dn

I d x da,

(7.40)

where

Tt(r, C) = I I,

I{ (w, z) K (o, C) L'@[ut](w) dudu.

This integral is also defined as the Cauchy principal value. In this step, wedifferentiated (7.3S) under the integral sign. To justify this procedure, we needto show that the integral in (7.a0) converges absolutely and uniformly withrespect to t. More precisely, putting H(r) = {z e H I lrl } r} for any positive

number r, we shall prove the integral

I(r)

r r ( r r | - ' l= [[ I I I l^(,, z\r2(z,oT4Vtr L'@l,;G)ldg,tl dxdv-

l l r o [ J / r1 " ; l L &J \ / ' ' )

converges uniformly to 0 as r --+ oo for all 1. By Lemma 4.2I, the above ?2 is

estimated as

llrv,t,,c)l' aea,t = u IIrII{(w,z)r"@[v2](w)P

alart

s c2tr2 [ [ Vr@,2)12 dedn,J J a

where c is a constant such that llL,@lu2]ll- s c for any t and l. Thus we have

tt g(,2)r2(z,ioldtd,tJ J H ( r )

/ r r \ 1 / z / t t ^ \ r / z1c* | I I lK(c,z)l ' dtd,n) ( t t [ i@,2)l2d'udu|

\ " / / 41 '1 1 \ l l n ' /

If z stays in a compact set of f1, the first factor on the right hand side tends

uniformly to 0 as r + oo, and the second factor remains bounded, which leads

to the desired conclusion.

Finally, by using again the fact that the linear operator ?' in Lemma 4.2I is

isometric on trz(C) , we can show that

xg at$ JI '(" '?'or) go Surddeu snonur?uor " sr (rn) t'rq leql eas an '

, sgos'r! - t't1

Eur11n4 '(s'lz);o Eurddeur snonurluoc e 4 Q)"t! lsrlt sraolloll!,gg.7 uorlrs-odo.r4 'tg't'tg = ,/ leql e?oN 'C yo Surddeur tb-s.'ril l€f,ruouer eql eq "'t! p.I

'*H)z '(z)(s), I

E)z 'O l=@)n,rt IH>z ,(r)(t)n

)

les a^r 'e ) s '3 11e lo3 'lxatr1

'(l'r) Jo Surddeur crlfleuelear e osle sr (^)rll Surddeu

esJe^ur slr l3rll saoqs rueJoeql Eurddeur esraAur eql 'o x H ) (l'z) lo Eurddeructlfpue-pe.r e u (z)rg' acurg '(1)a luercgeoc rur€rllag " s"q pu" ,g raldeq3 ;o(Z'g) fq peugap s.r q?HAr 'H

Jo ,I Surddeu leuroJuocrsenb e sacnpur o1 'uaqa

l=!'*H ) z '(4ta

,r Zi- = @)dte-fe '

,tq (J '.H)zV ) 6 lueuele ue eugep eM ,O 3 I fue .rog'CI x H > G'r) ol lcadse.r {lle ooC sssloJo sl gI', tueroaqlgogoord aql ur (z)rrfSurddeur l€ruroJuocrs"nb aql leql uollrasse eql;o;oord e ear3 aM 'A qrvuey

'? Jo uorlrunJ

crl,tpue-par e sr (3)Ifr1 qma leql aes etr snql'(J)J uo cr1,(1eue-l"er ar" lt pueb tt"'(t),t aceds ralpurq?lal eql uo cr1,(1eue-lear are sel€urproo? e{?rq acurs'tg aceds e{rlq eql uo suorlcunJ cglr(1eue-1ear are .t, pue ll ilt'6'g etuural ,(g

t=!'!7pyltp3="^,

e_6e urroJ aqt

ur uallrJiir sr rrrlaru uosseled-lralt aq1 3o '-o ruJoJ FluauepunJ eql '(g'3 ura.ro

-eq;) epu.ro; s,1rad1o11 uor; '1ae; u1 'cr1,(pue-lear sr 1lr1 qcea (eJourreqlrng'Qn'a'[1] srollqy

aas) I ;o uollcunJ-oop e sr (1)!fr1 qcee Ttsql ̂,roqs u"c a,u ';oord qql u! teql se

luaurnSre etues_eql Eurpadeg '1g sselc Jo e.rc !f,/ il* 'sr 1eq1 'snonurluoc ale

N/llqg'rlrQ/r{qe ile t"q} ees ar\{ 'g1'2 ure.roaq;;o;oord aq} tuoq 'I 1rout'ey

'gI'Z uraroeq;, ;o;oo.rd aq1 salalduroc srql

np x p (z )l'h) el ̂7 { or n O 1n)lt,t) 61,r (n' z ) ttr (4' n) y -t

frp xp (z)l,r,1l e),il { uO rO 17)( n} 61 ar (7, z)ta@, )) )r -t

.a)#="lnu'"il #-= .lilu'"[[#-=

a)#rulal I uossralad-Ira |t'l 902

206 7. Weil-Petersson Metric

t and s, then 1r1," is a conformal mapping of f'1(f1), because fi,, and .F1 have thesame Beltrami coefficient v(t) on -Fl. Thus, applying Cauchy's integral formulato h1,", we see that all derivatives of h1,r(u.,) with respect to tu are continuousfunctions of (w,t,s). Hence, all pa"rtial derivatives of f1,"(z) with respect to zand Z are continuous functions of (z,t,s) e H x D2.

On the other hand, the Corollary to Theorem 4.37 implies that for a fixedz, ft,,(z) is a holomorphic function of (t,s). Hence, applying Cauchy's integralformula to f1,"(z), we conclude that all partial derivatives of fl,"(z) with respectto t, s, z, and Z are continuous functions of (2,1, s) e I/ x D2. Thereforc, f1,r(z)is a C--mapping of (z , t ,s) € H x D2. In par t icu lar , f ' ( t ) = f t iz) is a C--m a p p i n g o f ( z , t ) € H x D .

7.3.3. Alternative Proof of Kiihlerity of the Weil-Petersson Metric

We shall give another proof of Theorem 7.15, which is also due to Ahlfors [6].Here, we use the fact that the first variation of the area element induced by thehyperbolic metric vanishes (Lemma 7.16 below), which is interesting by itself.

By a translation of the base point, it is suficient to prove the relations in(7.37) at the base point. We use the notation in $3.2.

We set

, l ' i ( t ) ( r )= e lu j ( t ) l ( f ' ( r ) ) { ( f ' ) " ( ' ) } ' , te D, z € H.

Then from (7.16), we have

,t,i1)e) = + [ [ Kk, 02-vl;) d€d,rt.r J J s

By an argument similar to that in the proof of Theorem 7.15, we see that ,/ti(t)(r)

is a C--function of (2,1). From formula (7.38), we get

Further, setting

t lhln1) =

J J "riQ)'t 'rQ)Q) dxdY.

pt(z) = (l(f)"(,)l ' - lU'),Q)l ') 'x'(f '(,)) ',

(7.4r)

(7.42)

we see that formula (7.3a) is rewritten in the form

hin(t) = t t vrcra$xg)e)o - l'(t-\')l')' a'av g.4r)

r r F P t ( z )

We can differentiate (7.43) with respect to 12 under the integral sign, becauseF' is a relatively compact set in -Il, and the integrand is a C--function of (z,t).Hence, we obtain

se uallrrtr sr ur3rro aql l" z uollcerrp eq1 ur "d Jo e^rlellJep aq? snqtr

'("!)l("1't'l - rrt(2:[-- (')'I) -

v-("|(,)'Gt)l - ,l(,)"(J)l) "((,),1)'v = (z)'�d

eA€rI etr 'uorlrugep eqt ,tB'raqunu

Ieer ll€urs flluatcgns {ue sr s 'ereg 'r"otr = "./ pue ln = rt 1as e11 ]foo.la,

.t-ft.....I = !,0='='\ffi

se{sz7os (a/'D ur pau{ap td uor?cunl eqJ 'g1''L BrnuraT

tr 'u"Irelqey $ d/t? feq|sarlo.rd qc-rqrh '0 -

t le ploq (Zg'f) ut suorl€ler eq1 ',trerlrqre ere / pue 'q 'f, ecurg

Uv't)

(ev'L)

(sv't)

'o = 6l !!,9 1 ,1tg

1aE er'r '(w'D pu" '(gt'Z) '(W't) uror; 'aro;araq;

.o - npxp ,_(z)HuG)16(,)' lUgrr*"ilel€q e,ll 'r1 o1 lcadsar qll/r{ |fi1 3o e,rrlerrrrep eq} 3ur4e1 '{l.rel.-,ols

.o- npry "-?)ny1,1t^(4' ldfu"lIurctqo a,ra '(gt'l) pue (W'2) ,tg

. n p x p " _(z), u @' ='

I v_Lr) w @g" | | = @ ffie^eq osl€ a,r 'r? o? lcedsa.r qll^{ (I7'z) Surlerlue.raglp '}xaN

pueq lq'rr aql uo urral puo'as aqr reql aas a^r ''.o1aq t;;Tffi:{?"i]J?i:l;

0,,, { l(#) ff}et,^r,t'^'il .(w t) npxp"-(z)u,{r"l'='l ffie)u+lzyralzf

'l&\"ll =

nt 7!e ,",!!qg

)rrlel I uossralad-Ira^Ut'/ L0z

208 ?. Weil-Petersson Metric

iQ)=kl,="r,,= fi t i @ -ian - 6!1 i, tz) +-i k))=-8Re{t+ a+\=-sRELa,{&\,

where /(z) = (0f"(z)/0s)1,=o.Thus we need to obtain more information on

f(,).First, we see the followings:

(i) i: = v = \rf 9 on H, whete 9.= 9i.(ii) / is continutus on C, /(0) = /(1) = 0, and lim,-- i1r1112 = O.

(iii) / takes real values on the real axis R.

For the proof of these assertions, we refer to Lemma 4.20 and Theorem 4.37.

See also Kra [A-58], Lemma 1.4 in Chapter IV, p'136.Now, we determine / as follows. Since rp is an element in A2(H,f), it fol-

lows that Xig " bounded on I1. Thus, integrating g three times, we get aholomorphic function rlt on H satisfying the following four conditions:

(u) ,lt"' - 9 on H.(b) / ir extended continuously to the real axis R, and satisfies that r/(0) =

/ (1) = o '(c) (Imz)r! '(z) -0 and (Im r)'t"Q) * 0 as lmz -' 0.

(d) {(z)/22 * 0, (Im z){t '(z)lz - 0, and (lmz)2t/,"(z) * 0 as z + @.

Solving the diflerential equation in (i), i.e.,

. ( t - ; \ 2 -

he)= _ftve1,

we obtain

ie)=-9+T16 -276;-a -@ *F(z) ,4 ' . . - t 2

Y t ' t 2 ' ^ \

where F is a holomorphic function on I{.

Let us determine F.. From_(ii), (b), and (c), we see that ]7 is extended con-

tinuously to R, and f = -rb/2 * F on R. F\rrther, by (iii) it follows that-rtr/2 + F = -{/2+ F on R. Thus h = F + rh/2 is a holomorphic function

on Il, which has a continuous extension such that h is real-valued on R. Hence,

Schwarz' reflection principle shows that h is extended holomorphically to C.

Then from (ii) and (d), this h should be a polynomial of degree at most 1.

Consequently, by (ii) and (b), we conclude that h - 0. This implies that

(7.48)

l=q,!

Gv't) 'tf lta 3 = 6)oun,fq

peugep sr / uorlcerrp eqt q d ry (1)oA arnloarw ,??NA eql.urrou lrun qlrrrr d 1erolf,el 1ue3ue1 crqdrouroloq e s! /,.e.1,I = o(tl,,,,tl ?eq1 qrns d ry W p (W)dlaceds lua3uel crqd.rouoloq e{t Jo luaruele ue eq d(r4g/d pr=j3 = A p.l

.lutrAatq 3_=

wtttyN

{q uerrr3 arc w71!A sJosuel NnlDaJn) uDtuuoutery aq1 ,.raq1lng

l=7"'ra3- lta

N

fq paugap ew !!A sJosuel ernlDarw ,?cNA aq;.

.*18 _-t4rzrtr'l'te ld

,(q uaar3erc atq,g srosuel arnlDarnc aql ueqtr .1lfq) o1 xul€ru esrelur aq1 sr ("rg) areq.u

t-u

, l?Q ,ffi*rrt J =r.lJ

^I

fq uaar3 ere zsp f,rrlau Jelqgx aqt qtl/ttPel€rcosse uor?f,euuoc )rrlatu aql Jo

7lJ qoqnrfs Ia,uolsrrqc eqJ 'y'/ uorsuaurp Jo/t/ ploJlrr"u xalduoc ts uo crrlau relr{gy p ae qlpt:ipIlqr=ci!Z,= rsp 7e1

'(y1 reldeq3 '[qq-V] nzruroN pue rqsefeqoy aas) splo;-rueu relqex uo f.r1auoa3 l€rluarasrp xalduroc Jo suorlou euros 11eca.r

,1s.rrg'elrle8au are crrleru uossraledlral\ aql Jo sarnle^Jnc leuorlces

ctqdroruoloq eql pue 'alnle,rlnc relsos eql 'se.rn1el.rnc rc?lg eql t?r{} ^4,or{s eA\

crrlatrAtr uossralad-Ila/$, aql Jo sarnlB^rnc .t.8.^l

tr ' 0 = 4 1eql apnlruoc am (9p.1) ruor; ,aro;alaqa 'f.reur3etur flarnd sr

. q(z-.2) o_ . ."(z-r) *=[ rk-r)\ rn"_(z)4, + (z),/,o Aro<ln" -

I -Gt-

I n "-

l"q? s^rolloJ lf 'acua11

' z - + - ?),frJ- - P),,fr-:--=- = (z)! (r)fi (r)fr t-,t.1-z_z \/',t "(z_r)

\-/..

602f,rrlal^l uossralad-Ira^{'9./

270 7. Weil-Petersson Metric

It is said that the Ricci curvature of ds2 is negatiae at p if Rp(V) ( 0 for any

direction 7. The holomorphic sectional cuntature KoV) at p with respect to V

is defined by ,u

Kp (v )= - D R i1mtu i l o tT . ( 7 .50 )j , k , l ,m - r

We say that the holomorphic sectional curvature of dsz is negatioe at p if

Ko(V) ( 0 for a"ny direction I/. The scalar curttaturv R is given by

N

R=Df t i i . ( 7 .51 )j = l

Theorem 7.L7. The Ricci curttatures, the scalar curtt&lure, and the holomorphic

sectional curaatures of the Weil-Pelersson metric are negatiae onT(f).

Proof.It is sufficient to verify this assertions at the base point of "(f).

We use the same notation as in $$3.1, 2, and 3. For simplicity, we set N =

3g-3.For a complex var iable z = n l iy ,wepf i do(z) - dndy. We wr i te

d .o (21 , . . . , 2n ) - i l o (21 ) . . . do (2 " ) f o r comp lex va r i ab les 21 , . . . , 2n .

Take an orthonormal basis {ei}l=rfor A2(f ,I l) with respect to the Peters-

son scalar product. Denote by ds21,yp = 2 Dilo=, \l$)ati d-tx the Weil-Petersson

metric defined in (7.35). Then, as was seen in the preceding subsection, it follows

that0h , t

h1t(0) = 6ik, f f {o)

= o ' i ,k , l - 1, . . . , N. (7.52)

Hence, from the definitions and (7.52), we obtain

, a 2 h -R i t ^ = _ f f i t u ,02 h , ,

Riltm = ffi(o), (7.53)

Ri.=-i#r,r t u 7 = ' '

at t = 0. Therefore, we only need to calculate 02hr7l AttAI^ at t = 0.

From (7.40), we have

1h;- r , . , 24 f f 01f i (21,2; ) ,- ^+ ( t l = - I . . . I - - - # I i { 21 ,72 )v iQ1)a (22 )do (22 ,21 ) ,Atr , r J Jptn dtl

where F is a relatively compact fundamental domain for f in 11, and (tz,rt)

runs over F x H. Furthermore, from (7.39) and (7'26), we obtain

\ x t?1 'a ) - - 1 [ [ * ,@r , zy ) I {1 (w1 ,2 - ) v2 (w1)do (w) . ( 7 .54 )

A* r JJu

salqerJ" uorle.r3alur eql ?eql ees elr 'suorleurro;su€rl snrqotr J I"aJ ol lcedsa.r qltmslerluera.Urp rurerlleg pue'()'z)y'()'r)X'@)op roJsalnr uorl€ruroJsu€r1 aqt,{B

'(32'�3@)>I (r2'�2)x Gn'�ilx (32")x (n'z)N (rz'�tm)x+

(gz'))x (z'2)x (2'@)x (n'z)N (s'r)v (rz'tm)>r+(?'r21, (9'))x (n'))x (rrl"'eol")x (z'z)y (3s'r)x- c

areq^.t(os'z)

'()'z'zm'rot'tz'rz)op(z*)*n (r*)tn (zr)cn (rz)tn C 'HxJ[

.. [ + =J JV6

6)v#e a'! zO

pug eA{ uaql '(lg'Z) Jo eprs pueq 1qEr.r eq} uo dr

Jo urlrra? Prlql eql u\ (rz'lll)X'(zz'|m)N o1 Pus'urra1 puoces eql ul (zz'rz)>I'(tm'cm)x o1 'rura1 lsrg eql u (zz'rn17o '(rz'em)N ol €lnuroJ srql ,(1dde a6

(es'r) 'H ) a'n '(r)op(s'r)x @'z)>r"il +- - @'n)>t

urelqo am 'ureroaql enplsar eql pu€ slnruroJ s(uaeJC Sursn 'a,ro11

(tg't).(z*,r*(zz(rz)ep(zm)urn (rn)tn ("r)rn (rz)ln g

"""[

[ #=

(r,ffi)=6)ffie^€q e/ll snql

'(9'@)x (rz'cn)x (9-"^1o Fz'tn)N+(72' rz) N (9'@) X (rm' zm) N (tz' tm) N *(zz'rz)>I (zz'tcrl)>I (rm'zrrl)>I (rz'zm)N= g

eraq^!

,(zm,rm(rz.z,z)op(zm)un (r^)tn ("r),h (rz)lrt g eHxrI

t * = @)#I I VZ,

.-. rltlzQ

plel,( (gq'Z) pue'(99'l)' (Vg' t) selnuroJ'acue11'(32'tr1ry (zz (h)tx (rz (rn)tx - y erar{^{

(gs'z) '(r*'rr'zz)op(^)tn ("r)'h (rz)la v "*"1

t #- = @;!*

(qq'r)

1aB er* snq;

.(zn)op (zn)t,t (g.,zn)to (rz,zm)ty . [ [ +-

- *!8JJ 1

- (zz'rz)tYg

aleq e \'uor1e1no1ea .repurts e fg

)rrlal{ uossraledlra A 't'l IIZ

02 h,t,

mintot.rA f I

= 2 I . . . I c v iQ)r {6 u{w) i^@}aoG,z,21,z2,wr ,w2) ,- o5 J Jr.r"

(7.60)where ( ranges over ,F.

We introduce the notation

_ t t -L1* ( ( ,2 ) = I . . . I K(q ,C) K(z1 ,wv) K(* r ,z )v1Q1) u* (wr ) do(21 ,w1) ,

J J H z

f f

L; r (C,z ) = I . . . I K( 'd ,0 K(a ,wz) K(* " ,2 )v1Q2) rn( .2 ) do(22 ,w2) .' J rH2 (z '61)

Then, from (7.60) we find

W,,

212

in (7.59) can be interchanged in an arbitraryas

direction 7 is given by

R'(v) = -xt27fo

where

7. Weil-Petersson Metric

manner. Thus (7.59) is rewritten

N

I Egr*i& ao11,'1i ,k-L

9.64)

h do(C, z) S 0,

='# | lr,u

Di,tm do((' z)' (7.62)

where

DiEm = Lin(e ,z1T-;qg + L1/(,4me + L1/C,2)

Since tr ir((,2) = L1,i(2,(), and since tr ir((, z) = L1*0(O,lGD i@

(7.63)

1tQ) for any 7 € f, we see from (7.62) that

0 2 h . , 1 , r t

W#lol = # J ...

Jr,, Eirm do(c,z),

where

EiErm = Q1/(,2) + L21((,2)lGr cA +Tffi) +2L1n(C,z)TRe.

Now, we show that the Ricci curvatures are negative at the base point. In

fact, from (7.49), (7.53), and (7.63), for any element V = Dl=rai (A/A$)o e

"o("(l-)) with unit norm, the Ricci curvature ,Rs(U) at the base point in the

21 1,,"t=t 1,,.

L^*(C, z) .

2t47. Weil-Petersson Metric

By the same argument as in the case of the Ricci curvature, we can showthat if KoU) = 0, then V = 0. Thus Ks(V) < 0, and hence the holomorphicsectional curvatures are negative at the base point. tr

Remark. Wolpert [253] obtained the following estimates for the curvatures of theWeil-Petersson metric:

(i) the holomorphic sectional curvatures and Ricci curvatures are boundedabove by -l/2r(g - 1), and

(ii) the scalar curvature is bounded above by -3(3S -\lar.

We also refer to Jost [A-49], Chapter 6; tomba [235], and Wolf [2aa].

7.3.5. Weil-Petersson Metric of the Teichmiiller Space of Genus L

We define a metric on the Teichmiiller space fi of genus 1 which corresponds tothe Weil-Petersson metric hsp on Q with S ?:2.

As wa.s seen in 52.2 of Chapter 1, the Teichmiiller space ?r is identified withthe upper half-plane I/. In fact, for every point r € //, we denote by I thelattice group generated by 1 and r. The torus R, = C I l, has a marki\g D,associated with the generators 1 and r. Then the above identification of H to

fi is given by the correspondence sending r to lR, Er).Let \lldzl2 be a metric on a torus fu = C/f, where r € H, and,\, is a

positive constant. Here, we impose a normalization condition on \zrldzl2 so thatthe area of .R, measured by this metric is 1, i.e., we put \, = l/tfirrn.

Now, for any I € C with sufficiently small lll, we consider a quasiconformal

mapping ft: Ro + Rt+t induced by a linear mapping

i,(z)=(t*_1=),+ - \2, z €c. ' \ r-r/ r-r

Since the Beltrami coefficient pt of it is equal to -tlQ - i +t), the derivativeof pr1 at r is given by

Ftil-1 p,=llliT=rl;;= r_i'

This p, is regarded as a holomorphic ta,ngent vector 0 / 0r on T1 at p = lR, , Erf ,and gives a basis for the tangent space ?p("r).

We define the scalar product of 0/0r and itself by

. ,a a. il 1 . 1nw p(

ar, a) =

I l r.t1, _ 11, )'i dxdy = all^ ry

This metric is the desired metric on fi, and is written as

r- 2 I t)-tzdsw p' = -2g^r1r9,l '

uo el€urprooc lts)ol e sr (z'4) 'ara11 '(1 reldeq3 Jo I'g$ ees) g Jo ern?onrls l€turoJ-uoc eql seuTurrelaP qllqr!\ u uo ,lzPlTd

- zsp 3lrlelu usruu€ruetl{ e erle; '1utod

es€q aql le (gr)J lo ((U)"f)".2 eceds lua3u€l eql replsuo? ol lualcgns s-I U'(U),2 "" rrrlaru uossrelad-llal\ eq1 saar3

@)W uo lcnpo.rd reuur I€rnleu e fq pe?npul (g,)Z "" clrletu e lsql aes aA\'.{1r1uepr eq1 o1 crdolouoq ere qcrqa JIastI o}uo Ur go susrqdrouroagrp Surlresard-uorleluerro IIe Jo (U)0//16, dno.I3 eql ,tq Ur uo s?Irletu ueluuetuelg Ie Jo (A)Weceds eqt yo eceds luarlonb " qtyla pagttuepr $ U aceJJns uuetualg pesolc e Jo(g); eceds rallnuq)reJ arll lsql ilres a^r '1 raldeq3 Jo A'g$ ur pelels se^r sy

crrlatr luossra+ad-Ila 4, aqt Jo uorlBlardrelul crr+auroa9 IBIluoro.SIC v

'g'g'Z

'ZZ 6 qlylr tJ

ro; g .raldeqC Jo g'8 tueroeql uI elnuroJ s,1.rad1o14 o1 spuodsarroc elil.uJoJ slqtr

' V P V W = d a n

el€q e.$ 'eroruraqlrng

. I ' Z 'E I J

. z - 6qll,lr rJ uo se?€urproor uaslarNleqrual o? puodse.r.roc q?lqr!\ 'Ig uo seleurproocser'r3 (B (/) uaql 'Vf

lW, - d las en g'"lzpl[y fq pa.rnseaur 'flazr.rlcadser '-r aU-'Io1 ur3rro eql tuorJ secu€+srp ete 7'7 l"q1 ees e^{'I 'Z'3U ul pe}e?Ipul se uaqJ

l lJdlf ,J-vt l^

JEU I

las eitr 'pueq reqlo eql uo

' tpy tp ey = dao

,tq uerrrS q ediltp Jo "^o turoJ leluaur€punJ

e o1 dn l? uo crrleru ere)urod aq? qtl^{ seprf,uro, qlq^taql 'rolreJ lu"lsuof,

> t c

It

:1,- "56rF. ,lt - - - - - - - - - - - - '

rrrlel I uossrelad-IralUt'l

216 7. Weil-Petersson Metric

R. The tangent epaceT = Ta"r(M(R)) of ll(n) at dsz consists of all symmetrictensors of degree 2 on J?. Every element c of 7 is written as

a=Adzdz*Bdz2+Bdz2,

where .A and B are smooth functions on U, A is real-valued, and B is complex-valued. This o corresponds to a real symmetric 2 x 2 matix

- rle+B+B i(B-B)l o=, | 4a - e1 A'- B -'Bl

Now, the inner product of two elements

ei = Aidzdz * Bidz2 +4 ilz2, j - I,2

in 7 is defined by

(or,azln= [[ t(dfi2)p2 dxdy.JJR

Here, tr(d1&2) is the trace of the matrix d1d2 with respect to the metric dsz,that is,

tr(&fi2) - AtA2 + 2(Bt6 +E[a) '2p4

Thus we have

(ot, ozl n = i l LlA,,q, + 2(B1g ar,,nr)l ry

Note that the following two types of elements in 7 correspond to zero vectorsin [("(E)):

(i) A vector induced by deformation of the scale Lactor p. This is an infinites-imal deformation r|,p2ldzlz which is generated by a l-parameter family

{p'"'$ld"l'}re;- of conformal deformations of the metric ds2, where ry' isa real-valued function on ft.

(ii) A vector induced by diffeomorphisms of R to itself. Namely, this is an in-finitesimal deformation

o' (* 4rz * oa '-'\

' \dz

- Aro'- )

induced by {/i(ds2)hen of deformations of ds2, where {.fr}ren is a L-parameter family of transformations of .R which is generated by a vectorfield X = a(z)(0/02) on E.

We shall obtain a condition on an element c = Adzd2 * Bdzz + Bdzz suchthat c is orthogonal to all elements of types (i) and (ii) in 7 with respect to thegiven inner product (., .)n. First, in order that o satisfies

(,, g\ a =; I LA{t

dtdy = o

lurodalar,r eql urorJ parpnls ere saceds raflnuq?ra; araq,rl'[776] 1o7y1 pue'[996]equro{,1 '[62] equrorl pus reqsrd'[lg] arreuraT pue sllag,[6t-y] tsol o1 ra;ara,ra. 'aroturaqFn{ '[0gz] prrc '[6gu] '[976] r.radlo1t osl€ aas 'snlncle? sse€I4l eqluo pas€q sl WH,!r '[996] fredlo1t ,(q uarr€ sr;oo.rd a^r]€urall€ uy '[/] pue [g]sroJIqV ot enp er" rrrlaru uos$eledlre \ aql Jo flrralqey .ro; .raldeqc slqt ulsyoord o,lrl'[g]U] IIaAI /tq pecnporlur lsrg ssl{ )lrleur uossreladlre1\ eqtr

'[gg1] epe4eN pue'h9I] r{eznsteW'[ezr] tllselt Pue ery '[Oet] *ty '[28] "tx PIr" raulPr€g aas 'sdnor3 ueuralygo flqrqels IsuroJuoers"nb o1 ,(Solouoqoe ralqcrg erll Jo suorlecqdde rod '[gg-y] "ry ,tq looq aql ur punoJ sr ,iSolouroqoc relq?rg eql Jo leep 1ea.r3 y

'(gsr-qqrdd'g le1deq3 ul U I$ '[tfV] aas) a1.reg ot enp sl Z$ ul;oo.rd rng '[6-y] sroylqy urg .ra1deq3 Jo I "uuraT aas '(9'2 euwal) €tutuel s(rellnuqtrel;o;oord e .rog

'[tlt] ptt [971] olounselI pue'[lzt] ttx fq e.re saues ereculoduo sraded ,tueur;o auog '[gg-y] .rauqal pus '[89-v] e.ry fq s{ooq aq} w peulet-uoc are 1$ ur sarras erecurod pue $uroJ arqdrourolne Jo slretep a1a1duro3

'ltt-v| €rnlpox pue,$,orro4 pue'Lg-V] srlepox'[qq-V] nzruoN pue rqsefeqoy '[69-v] srrreg pue sq]lgrr9 fq slooq aq] m punoJsr ,trlauroe3 l€rlueragrp xelduroc uo lerrel"ru .,t.ro1cnpor1ur 1n;d1aq;o leap 1ee.r3 y

saloN

'(U)Z uo daf rrrleur ueruueruarguossrelad-llel\ eq1 sa,rtE qeH,u ((U)J)oJ uo euo eql qlr^\ sapllutoc lcnpordrauul s-Iql l€ql aas ern'.sp se U uo crrlel'u cqoq.radfq eq1 3q1et'relnctlred u1

-d af f

4f*pg'fr ll duz = a(za'tol

fq ua,rrE sr '-6,

ul (U'I = f) !,fi + !4t = fio slueurela o^rl Jo lcnpord reurn eq? 'rarroe.ro;41 '1urod

espq eqt le (31r)J l" ((U)Z),,2 aceds lua3uel arqdroruoloq aql ot spuodsarroc 1,lo {(U)zV ) $ | f +,fu} = o;, aaedsqns aqt ?eqt s^{oqs uolt"^rasqo srql

'(A)zV ur ,f euros rc1 rlt * fi = o s€ uellrrrlr sr o ;r fluo pue;r(g) p* (r) sad,t1 Jo sluauala 11e o1 leuoSoqlro $ L ul p luatuela ue 'acuag

'u uo EerlueJeJ-Ip crlerpsnb crqd.rotuoloqJo (U)zy aceds aql o1 s3uoleq ezpg snq; 'crqdrour

-oloq fl g 's! t€ql

'0 = Zg/gg leql aas aar 'frerlrq.re x (zg/g)(z)o - y acurg

's - npxp" #"[ I + nPxP "

#"[ |e^"q e!$ slmuroJ s(uearc urog '(g) ad{1 ;o 6/ luaurala ,tue .ro3

o - n w P

sagsrl"s la lsrll repro ur 'lxeN '0 eq plnoqs y '(r) adfl;o 5/ fue ro3

(At.#r)"i l =a(d'ol

Lr7, saloN

lurod,uarr\ aql urorJ perpnts are seceds ra[nuqrreJ a.raqaa '[776] 1o1q pue '[996]

€quro{L '[62] equoqtr pus raqsrJ 'Lg] "r,r*"f pu€ slleg '[6?-V] tsol ol .raya.raar'arourraqlrn{'[0gZ] pue'[6gU] '[gy6] 1rad1orlA osl" eas'snFrlec sse€I4l eqluo paseq sr qerr{^r '[996] lradloA\ fq uer€ sr ;oo.rd e^rleuralle uy '[/] pue [g]sroJltly ol enp are Jrrlaur uossJaled-lre11t eqt Jo fluelqey ro; raldeqc srql urs;oord orlrl '[g?Z]

IIel\ ,tq pecnporlur lsrg se^t crr]eru uossraledlre1\ eql'[gg1] epeqeN pu€ 'hgI] rlnznslelq

'[tzt] tl{.eru Pue erx '[0at] ttx '[48] tty Pue reurPreg aas 'sdnor3 uetutely

;o flqtqels leuroJuocrs€nb o1 fSolouoqoc ralqrrg eql Jo suorlecqdde roJ '[89-y] erx ,iq looq aqt ur punoJ sr ,(Soloruoqoc re1lrrg arll Jo leap 1ea.rE y

'(ggr-ggrdd'g raldeqg ul ZI$ '[tf-V] aes) a1.reg ol enp q A$ ul goord .rng '[6-y] srolpy urg raldeq3 Jo I "uureT eas '(g'2 euural) srutual s.ra1nuqcral;o;oo.rd e .rog

'[tlt] pt" [971i olounseltl Pu"'Fzt] ttx,tq ale salras er€culoduo s.raded fueur;o auog '[99-y] rauqe1 pu€ '[89-y] e.ry fq s{ooq aq} ur peulet-uoc ers 1$ ur sarras ?J€?urod pus surroJ crqdrourolne Jo slretep a1a1duro3

'[tt-vl errepox pue ir,!,orrotr{ pue'[Zg-v] errepox'[qq-v] nzruoN pue rqse,leqoy '[69-v] srrreg pue sqlgrrC ,(q s4ooq eql ur punoJsr f.rlatuoeS lerlueresrp xalduroc uo lerreleru i{.ro1cnpor1ur lnydleq;o leep 1ea.r3 y

seloN

uosslarad-rra^\ aq1 saarE q?Hi$ ( (u )rl, "" (""Ji

Jr; #i""$;"trjiT"T:'#rauul slql leql aes em'rsp se U uo tlrlatu cqoqredfq aq1 3uqe1 'relncryed u1

"d af f .ffi"L'fi JJrz=u(zo,ral,(q uarrr3 sr !

q (Z'I = !) !,1, + !fr = fra sluaurala o/rt Jo lcnpo.rd raum aql '.ra,roe.rotr11 '1urod

es€q eq? fe (U)'f l" ((U)"f),,-f aaeds luaEuel arqd.rouroloq eql o1 spuodsarroc ,1,fo {(ff)zy ) ,1, I f + fi} = } ecedsqns aql }eqt s^otls uotl€^resqo sIqI

'(a)cv ur 4l auros ro3 ril * fr = a se uallrr^r sl ̂o JI fluo pue;r(l) p* (r) sed,(1 Jo sluetuele 11e o1 puoSorllro sr -L u\ p luauela ue 'arua11

'u uo slerluereJ-yrp crlerpenb crqdroruolorlJo (U)uy aaeds aq1 o1 s3uolaq zzpg E\qL 'ctqd.rour-oloq $ g 'sr 1eq1 '0 = zg/ge l€rll ees aar 'frerlrqre il (zg/d@)D - X eculs

's - npap t t*" 1 [ + npop " t*"

I Iel"q aar slmuroJ s(uaerC uro.rl '(g) adfl;o g/ luaurala fue ro;

o- np,p (#r.#")"ll =a(s,o,)segsrles la lsql rapro ur 'lxaN '0 aq plnoqs y '(r) ed,t1 3o 5/ fue .ro;

LtzseloN

2I8 7. Weil-Petersson Metric

of ha^rmonic maps. Moreover, see Takhtadzhyan [219] and [220], and Zograf andTakhtadzhyan 12641, [265], and [266]. tomba [235] showed that the sectionalcurvatures of the Weil-Petersson metric are also negative. It is also known thatthe Weil-Petersson metric is non-complete; proofs are found in Chu [49], Masur

[143], and Wolpert [245].For the subject in $3.6, we refer to Fischer and tomba [72]. Such a diferential

geometric interpretation of the Weil-Petersson metric is closely related to thePolyakov integral in string theory (see, for example, Polyakov [176]). See alsoNag and Verjovsky [164].

For the Weil-Petersson geometry on moduli spaces of higher dimensionalcomplex manifolds, we refer to Besse [A-17], Fujiki and Schumacher [77], Koiso

[118], [119], Schumacher [190], and Siu [208].

'ttr*9rrart>oo-Iu>r)="7r4

se cAA ssardxa u€) ell\ 'Z/u ) od > 0qq^{ 0, elqelrns e rod'01 lo

oLV sr)rc eql rt,"iuor qrlqi{ (J o1 lcadsa.r qlrm)

H uo 3A4Jo 1JII e eq, cA Ial'C sraAor pu€ J o1 s3uolaq (t < V) zy - (z)oL

leqt pue '{pp} -._i' ;o fuaurela atuos go lurod paxg e sl I }eq} etunsse detueM 'fI eueld-g1eq reddn aq1 uo 3ur1oe A Jo J Iapour uersq)nd e ar1e1 '1srtg

'cA1 - A uo p? o1 pnba sr pue (rl4

ul I /tq g 3uo1e ,,Eur1sra,r1,, aql sluasardar q?rqa g,;o Surdderu l€ruroJuocrsenb e

lcnrlsuoc e^r uaql ', Jo pooqroqqErau palreuuoc flqnop e ''a'r 'pooqroqqErau

r€lnqnl € seruoceq cAA Wrn llprus os p luelsuoc aarlrsod e esooqc ern 'era11

'{, > (C'd)d lA ) d} = ctr4

les 'ltsp fq pacnpur acuelsrp aql 'U uo ecu€lsrp cqoq.red.ilq aq1 eq d 1e1'sEurddeur

leuroJuocrsenb Sursn fq uorleur.rogep e qens luese.rdar e11'I'g'3ld pue g raldeqCJo U$ ees'sraproq eq? SulnlSar fq uaql pue

? r{13ua1 cqoq.radfq fq Eurlsranl fq 'p 3uo1e g' 3ur11nc {q peurelqo Q. saceyrnsuueruerll pa{r€urJo {U > I I tg} fguey aql sueew 'uo ataq rnoquorlDulto{ep Ntreql pefiec fldurrs sr q)rq^r 'p o1 lcedser q]l/{ A lo uoNlDu.totep uesptTlleyouetreqt t€rll ilerdg

'{sp cr.rlaur crloq.redfq eql ol lcadsar qrlta C crsapoe3 pesol?aldurrs paluarro ue x-r.{ '(Z

<) f snue3 Jo er€Jrns uuetuarg pesolc € eq U larl

suor+errrroJec uaslarN-Iarlrued'I'8

'seleurProoc ueslerNleq?uag ,{q uroJ l"}uau€punJ uossJaled{aM eqt Jo uol}s}-uasardar alduns e 'flaureu 'e1nur.ro; s,1.rad1o14 e a,rord aiu 'g uotloeg ur ',{11eutg

'fu ur uotleuuoJep uaslalN{eqtueJ e fq paurur-Jalap Jolce^ lua3uel eql etelnrpc en'4 uollcas ur '1xag 'sEurddeur

l€ruroJuoc-rsenb Sursn fq suorleuroJep uaslerNlaq?ueJ eqrJtsep am '1 uorlcag ur '1srtg

'salsurProoc

ueslarN{aqrueg Eursn fq (Z ? 6) 6a uo uroJ leluaruepunJ uossre}adjla1\ eq} Jo'1red1o11 'S ol enp 'uorleluasardeJ

InJlln€aq e a.,rr3 IFqs e^\ 'raldeqa sql uI

JrJlatr tr uossJa+ad-lla/y\ aql PUB suol+BIIIroJa( uaslalN-lallruad

g ra+deqc

220 8. Fenchel-Nielsen Deformations and the Weil-Petersson Metric

VvN deto'*ation;t

*,,-'-H*..**--/

Vl _ Li+ ll

\qc mapping

\

F ig .8 .1 .

Next, for every t € R, we define a quasiconformal mapping ut of 11 ontoitself by

Here, d = a;tgzt and e = -t/(2eo). This tor gives a surgery of I/ along the axisAro. Note that the sign of I is different from that in Wolpert's papers [247] and

[251]. (See Fig. 8.2.)Now, denote by 4 the complex dilatation of urt. A simple computation gives

r1 (z ) - f7x ,@i , z€H.

Ilere, 17 is the characteristic function of -I = [" /2 - 0s, T f 2 * d6] on R. Further-ITlor€, 11 satisfies

rt o.yo . (T'o/t0 = ,r.

Thus, 4 is a Beltrami coefficient with respect to the cyclic group (7e).On the other hand, it is clear that the set ,l-c consisting of all elements in l-

which cover C is { Totool - r l t € f } .By l i f t ing th is FN deformat ion to. I l ,we have a family of self-mappings of I/ which give surgeries along the axes of allelements in l-c. Thus, we can construct a family of quasiconformal self-mappingsof ^t{ which induces this FN deformation as follows. Denote bv (zo) \ l- the setof all right cosets of l- with respect to (to), and set

( z , 0 < d < t - e o* ' ( r ) -

{ z e x n ( r ( 0 - t + e o ) ) , , - 0 o 1 g < t + 0 0

I z e x p ( 2 e d e ) , t + e o < 0 < r .

'(o='lJg-\ "P = 'a\ l,,r,g / ,p

o1 lenba sl (J)sJgo lurod es€q aql 1e {U 3 ll(rrt)d e rnc eqtJorolral lua3uel aql l€rll apnlcuoc u?, a^r'61'9 ureroeq;;o;oord aq1 ur se'uaq;'9 .reldeqg Jo I'I$ uI peugap sl'd(n eraq,n "dm = tot las en 'rrl drerra rog

'ploJrutsru leer e s€ pereprsuor (.t)sJ ur uor?euroJap NJ slq? Jo rolcel

lua3uel aq1 alnduroc IFqs ar* 'lxeN 'C 8uo1e g Jo uorleuroJep Ng eq1 sluasardert{cr{rlr 'r(._;r'H)g ul {U > I | ,t/} "nt.,t e peul€lqo aAsq a.{r 'relncrlred u1

'909

-ggg'dd '[1y6] tradloA\ aas 's1te1ap erotu roJ 'Surddeur 3ut11nsar eq] azll"turouueqt Pue

'J \ (0.t) 3 r-1,(01,) {.rarra .ro3 eroJeq PeqlnsaP se (c1-)L uo 1 fq

(",V)L 3uo1e ,tlartrlonpur ((lsrrhl,, 'i(11en1cy 'ff ul t fq o'V 3uo1e ,,s1srall,, ql.rq^r

'ror Sursn fq rtlpcrrleruoa3 pue fllle{p eroru,,rrn lcnrlsuof, u€e aJ1yIJDuu�e[

'g 3uo1e Ur Jo uorleuroJep

N1I eqt sluaserdar qrlqa (J)J u! {U f I | [,7rrr]] ,(lTueJ s ul"tqo eirl 'snq;'Q)l > ['arn] lurod € seunuralap pue 'slstxe

H lo ,an Surddeur cb-rr/ Iecruouecaq1 '3 fraaa rog 'aoue11 '1(,1 'U)g o1 s3uolaq trl ltsql uorlrugep eql urog reelc st lI

, J\(oL) )rL(tort\1 7 -4t

' !-l

'z'8'ttJ

,,rrT

tzzsuorl"ruroJeo uaslerNlaqf,uaJ' I'8

222 8. Fenchel-Nielsen Deformations and the Weil-Petersson Metric

(Here, recall that such a tangent vector is considered as an element of A2(H* , f).)Furthermore, Theorem 4.37 gives an integral formula for ti = (0q/0t)11=s

as follows. We set

where

It is easy to see that

Thus we obtain

x , , zuslz) = --yl largz)=.'lvo z

]13tt? -/ l l- = o'

i t(z)=-+ ilr,G)ffid€dr, ze c. (8.2)

As has been stated in the proof of Lemma 7.16, we know that

(i) (d)" = L, on C in the sense of distribution, and(ii) d(0) = tt(1) = 0, and b(z) = o(lzl2) as z -----' oo.

Conversely, these conditions (i) and (ii) characterize tir in the class of contin-uous functions on C, which can be easily shown by using Weyl's lemma (Lemma4 .6 ) .

Now, to get a simpler representation of gc, we rewrite formula (8.2) asfollows.

Lemrna 8.L. The deriuatiae b is written as

( l a

i(z) = -, \1,*" # o, * L*^r,) *.,.,,,.,F,,

rhotr,(,) (8.8)

for eoery z € C. Here, argz lakes aalues in l-r,r), and

F.,(z) = -#U,*",'' o# o, + fibs1et\ * fir.,1,1,wherc P.r(z) is a polynomial of degrve aI most two. This P, is aniquely deter-mined by the conditions that Fr(O) = 4(1) = 0 and that Fr(z) - o(lzl2) asz + @ .

Moreooer, the series on the right hand sid,e of (8.3) conaerges locally uni-

formly on C.

Prool. First, substitute the right hand side of (8.1) for p in (8.2). Then we have

(q's) '*rr uo)17

!

serrr3 1'g evrute1 'loo.td

'*H uo fryu.r,ofiun fipoao1 safitaauoc ycnlm

,,. J\(oL) )L,lL\ v(_cdt

"\r) I ?7oq7 snollot 7t

'sacuoTsrunc.nc 6uto6atol eql repull 'Z'8 uraroaql

E .s^rolloJ uorlresse aql

uaqa'r,61 L4l- s€ e?uereglp slql ?as a1yo^rl lsoru 1e ear3epSo leurou,tlod e st

acueraJrp slql teqt saqdun qctqn 'oo + z w (rlrDO sr pue 'g uo ctqdrouroloq st

( o),r. ,tf:_\_(,),1ioplpQ'))a X; (o)r)oal

"'-g),f,"" tlr J Q)Lo1

l?q? u/rroqs sI 1I snqJ

'J ) L',L' (Loon) - t(L" t)

e^eq eir{ 'uorlnqr.rlsrp Jo esues aq1 ur 0n = z7 acurs '1xe11

. { reoto1' *,n ooz

T\ ' - -t !

'YGYx"r*t I'--

tP {z?o1- (3)({"e"''3}xeru's)x '?t?,} ffi ""1

":=

'yor=) 'tp{. =tr _"[}ffi "'[ ":=

onz ," :2(r -)) ,

[ [ ", = (,), t'p?p666ffi,xt(t_4,

JJ ?eleq ad\ ueql '(?'8)

Jo aprs pueq lq3rr eql uo urrel lsrg aqt (z)1 ,(q aloueq'9 uo fprro;tun

f11eco1 pue flelnlosqe sa3raluoc (f'g) f" eprs pu"q 1q3u eql uo selres aq1 'fpealC

. (z . )XI . ))) = G,))a G-z)tles at\'eJeH

<oL>r/".J\<ot> tL

/ t r J\(o[]l't

(t - ,r"r;) 3

't plp (z'))a 611&(())r)0z (r'e)

"fl +ttplp (z

'[[:--(z\m

IJ I "' '))uO)0"

tzzsuorl?ruroJao ueslerN-Iaq)uad't'8

224 8. Fenchel-Nielsen Deformations and the Weil-Petersson Metric

where P;7 = S.On the other hand, a direct computation gives the following, Bol's equation:

/ t \ ' ' / r ' \ 2

1.7'"ttl = - (i,lThus, differentiating both sides of (8.5) three times, we have the assertion. tr

In general, for any simple closed geodesic C on r? and any element T e fwhich covers C, we can construct a similar basic series as in Theorem 8.2. Moreprecisely, let o and 6 be the two real fixed points of 7s, and set

(a - b \2/ \ - --- l - ---------- -- ' t o \ . t - e _ a ) 2 ( z _ b ) 2 .

Theno c = ( r r , o t ) ' ( t ) '

converges locally uniformly on .[/, and belongs to A2(H,l-). We call this Oc thePetersson series for C. Using this series, Theorem 8.2 is reformulated as follows:

Corollary. Let N(l) be as in 52.1 of Chapter 7. Then

i . _ r -u = --) f i 'Oc mod N(f) .

Prool. Using the notation in $2 of Chapter 7, Theorem 8.2 and (7.15) in Chapter7 imply that glvl = *Oc. Hence, (7.17) in Chapter 7 gives

Hlul= -*^o'U;

Thus, the assertion follows by Theorem 7.7.

8.2. A Variational Formula for Geodesic LengthFunctions

Fix a simple closed geodesic C on ,R arbitrarily. For every point p = [^9, /] €?(.R), let Co be the simple closed geodesic on ,S freely homotopic to /(C), and

denote bV tc(p) the hyperbolic length of Co. Recall that tc is a real-analyticfunction on T(,R) (see Remark 1 in $3.2 of Chapter 7).

Here, we compute the variation of 16 at the base point po = lR,fd]. Moreprecisely, take p € B(H,f ) arbitrarily, and let wtp be the canonical tpr-qcmapping of f/ for every sufficiently small real l. Then rptP determines a point,

say pr, of T(.R) for such t. Under these circumstances, we compute the value

tre ( 'vo) \ r

o

( z zY 1 *=",-, !'tnoa17t

uotu \(-*t,o,

- I*fut) ",v j ())" I L-

= (rt)"d(c7p)

uretqo el,l.'(z)n - (tU)" acurg '{og ) z

l r\ = !.q pu€ {y > lrl > ll H ) z} = og sureruop eqt uo asoqt se qer3alul aq}

alrraer pue '(/'8) Jo aprs pueq lqErr eql ,(q ,t1r1enba slql ul ,14 aceldar ',uo11

'o+z'&- dn=(o)ffi

- @)"d(cg):"lnuroJ freurt.rd eq? e^€II au 'ecue11

.?D 'Q)4tu +'(o)\p: = ?u)41

1€ql s^\oqs (9'8) 'pr"q reqlo aqt uo

cf f v 't'p?p (z'))uQ)n I I ;-

= @)4

:uorleluesarda.r p.r3alut eql seq

'c>z ,L121 rhi=@4t

1eq1 sa11dur1 zt'? ueroeqJ 'fl

vo nrt = rltn l?r{t 1'aoDI e^{ ueIIJ'9;oSurddewtb-rt| I??Iuouec aql eq nJ +e'I

'*H ) z '(z)rt

lrfz '0

H ) z '(r)rt

(z'e)

(g'a)

1es aal'1xa11

'r1Eo1 - (d)ot

l€r{t slllolloJ l\'c7 1o uor}IusaP eq} /tg

'zty=Q)r_Gr^)ooLo ttn

1€rl1 uortrPuoc eql ,tq paurur.raleP q (I () r1 luelsuoc e '1 .r(laaa

ro3 'uaq; 'g sra^oe (t < V) zy = (z)oL 1"ql eunss? feur a,t 'I$ ul sY 'loo.t4

a/ )L \ ' | "Oj,rt I tU = (d)oaQ7p)\ 6 / 'g'Suraroaqr

'([62] rautpr"C 'Jc) u,laourl-ila,ll fletluassa s! sIqI

o=ll lrt.

l(rqcti = Qt)od(c7p) IP

suorlounJ {ttua1 f,rsepoeC ro} "FruroJ [uoIl"-Ir"A V 'Z'8

9ZZ

226 8. Fenchel-Nielsen Deformations and the Weil-Petersson Metric

Furthermore, since

)"ft(c,4=-+(+-*)we see that

i ^ r " {n ( . l i ( , r z ) -R( \c , z ) } = -1 i { . , 1 " _ . ) : . ' }, ? * | l , z ) C , ? * L ) " ( - , s n - t ( - z )

=-i^u*#=-bHence. we have

Finally, divide F's into domains {Z(f) | I e f}, where F is a fundamentaldomain for l-. Then we conclude that

I 1,"8*'o =,,.H,,. I l,urct (#)" oro,= [[ u{0 ,H,"(#)'0,0,=i,,2",."

This completes the proof of Theorem 8.3. D

(dtc)o"oi = - + t | ̂ ,,,"ret (-!) ut,

=+11."(g.F)**=?Ru 11,"8**

8.3. Wolpertts Formula

We have computed the tangent vector (represented by z in $8.1) of the FNdeformation with respect to C at the base point, where C is a given simple closedgeodesic on ,R. By a translation of the base point, we ca.n compute the tangentvector field on (the real manifold) ?(r?) associated with the FN deformationwith respect to C, which we denote by 0f 7rs, and call the Flf ueclor fieldforC. Namely, 0/7rs is the vector field obtained by applying the FN deformationwith respect to C with unit speed with respect to the hyperbolic length. Notethat 0/016 is a real-analytic vector field.

.'c!8 - = ('"t9,3\ dtao- -cte \B e/

(%-,Y.) "rn,, = (::g\ (,4,\- cto\s I / \8 /'"" ,c78

el€q e^r 'fre11o.ro3 Euro3ero; eq1 ,Lg .too.r.4

.,c!g _ = "rg

clg ,clg

'A uo ,C puo C sctsepoa| pasola a1dtuts IIo Jo[ .g.g uorlrsodo.r4

'ur?roeql fr,7tco.r,dtca.t 3ur,uo1oy eql sl t.g ueroaqJ o1 zt.re11o.roc reqlouv

.(.)ctp=(.,"19-\*^.\ s/'zt.re11orog

tr 'uollrasse aql e^eq ea\ ecuaH '(rt)"o(tlp)

o1 lenba sr aprs pueq lqErr eql l3rll sa,rr3 g'3 ueroeqr 'pueq .req10 aql uo'(t'H)gU 3 r/ frana .ro;

" ( """.,t) "g 6=\[/

( , ,t",-"v i) d htu e1z-

\ -" I/

(, '!#r\ d^ qeaz= (,1 ,'"-n r\ "^u\ s / \ 9'l

eleq e^r'61'2elouwe1 pue Z'g r.ueroer{J o1 ,(re11o.ro3 aq1 uro.r;,1s.rrg'(U)Z.lo od lulod eseq eqt te

flrlenba JeruroJ eql ^{oqs ol seclsns 1r '1urod es"q aqlJo uorlelsu€t1 e ,Lg.too"t4

.(. )orp = ( .,'"-e-r\ " ̂6\ g '/

,.a.?(d/vr6 oI Tcadsa.t qpn pnp aq7 6uu1o7 fiq uaa$.toTnrado eq? il * puo '(6'6'7$'lc)

(g)a pyotguou p^t eql- uo etnl.rnrls ralilutoc lsouqo lDrnpu ?Tl suDeu c ?r?qn

'crp= (%r\.\ I

'/

'A uo C ctsapoaf paso1c alilutts fr.r.aaa Jo,{ .V.g ruaroatll

'urero?rll fi.4t1onp Suruolo; eql ^roqs a1il'clp ppg rolcal 1ue3ue1oc crlfleue-par eq? U.g$

ur peugep aleq ea'cl uorlcunJ {fue1 crsapoe3 eq} urorJ ,pueq reqlo eq} uO

"InuroJ s,1rad1o11'g'g

Finally, we have the following, Wolperl's fonnula, which is proved later'

Theorem 8.6. Fb a systern of decomposing euraes L = {Ci}?c=13 on R arbi-trar i ly. Denote by { tcr, . . . , lcsg-s,0cr," ' ,0c"o-"} the Fenchel-Nielsen coordi-nates associated with L. Set rs, = (tgrf2T)|st for eaery j. Then

3g-3

u w p = D O r " o A d l 6 ,i = L

Corollary. For eaerg simple closed geodesic C on R,

l a \, * , I fu, ' )= -drc( ' ) .

Proof. Take a system of decomposing curves which contains C, and apply The-

orem 8.6.

Now, to prove Theorem 8.6, we take a base of tangent vector fields

{X,, . . . ,Xu'-u} ={#,, ,#,#., ,#;}on ?(r?), where ?(.R) is considered as a real manifold. Further, we set

{ t r , ' . . , o o g - o } = { t c r r " ' , t c " " - " , T C r , " ' , T c " " - " } .

Then arpp is written in the form

a;idx; A dxi.

8. Fenchel-Nielsen Deformations and the Weil-Petersson Metric

This shows the assertion.

uwp =I l j <i<6e- 6

First, we shall show that every a;i is invariant under the FN deformations

with respect to C3 for every /c.

Lemma 8.7. For eaery i, i , and k,

- 0 on r@).

Proof. To prove the assertion, we use some basic notions and results from the

differential geometry. See for instance Matsushima [A-72].Let /(X) be the interior product with respect to X. Then the corollary to

Theorem 8.4 gives

Surddeur l")ruousc crqdrouroloq-llue u" auuae'aue1d-g1eq reaol aql sr *lT areqa\ 'S

Jo sJ / *H aSerur ronrur eqt ,S ,tq alouaq .g

uo 3ur1ae S'Jo Iepour uersrlcnd " aq sJ lel '(U)J > [/

,S] lurod f.reaa roJ 's,lrolloJ

s€ Jlastr oluo (g)g 3o ' t fus'Surddeur e secnput / ueql 'f uorlcegar e s?turp" U

1eq1 flrlerauaE;o ssol lnoqlr^a erunsse feur ell 'uorssncsrp snll Jo tser eql uI('flrxalduroc ,(lesseceuun pro^e o? 'slurod pexg Jo sles eql go lred etuos

lTrrro allr araq,ra 'g'g '3rg aag) 'W uo scrsapoaS pesop aldurrs Jo raqunu alruge Jo sl$suoc d leql etoN 'f

;o slurod pexs Jo 1as e{} sr Jr eJeq^i!, ,ry fre,ra .ro;rtg - 14 Ud l"q1 pue'oaa1 raproJo og;o Eurdd€ur-Jles l€ruroJuoe-rlue ue'0U

Jo 1 uorloeuer e Jo uorlcrJlsar e saruo?eq Y uot?ceUar q?€e leql ees el!l' ueqJ'Ieqlo

qcee qlr^{ }uepnuro? ale (6'1 = il fl uo 7'![ Jo lI v''rrg slurod pexgJo sles aqlueql'/ r{?Be roJ r'!4 p uorlcager eq} aq (A'I = il llf pue'!7 Suop luacetpef11en1nu slued aq z'!4 pue I'id fal

'J ) lI fra,,re .ro; :uor?rpuoc 3ur,r,ro11o; eq1Surfgsrles OUr er"Jrns uusruarg e pug oiu '(l >) f'r7 frena JoJ uorlsr.uJoJap Ndaq1 Surfldde 'snq;'q13ue1 cgoq.radfq eru€s eq? qlra\ scrsepoa3 o,lr1 olur({g;o(t'Z'l = t) f't7 fes 'luauoduroe frepunoq qcse sepr^rp {1 ;o slurod paxg;o {rg

1as a{} '.rarroatotr11 '(g'g ura.roaq; o1 frelloroC aqt 'Jc) {d go f uorlcagar aqt s€r{{2, slued ;o rrcd f.re,ra ueq; '7 uelrS aql o} Eurpuodsa.rror Ur Jo uorlrsodtuocepslued eq1 "q

"51{'dl1 = dpl

'1ce; u1 'uorloegar l€ruroJuo?-rlue ue Surllrurpe

Qr ue aleraua3 uec an 'suorleurJoJap NJ elqelrns rage 'pueq Jar{?o aqt uO

'pa3ueqaun srlapy txpllp3- "^, uolleluasarde.r eq1 'lg {ue o1 lcadsar qll,lr uo-r}?urroJap

NJ ar{l Sutfldde fq og' raqlou" o} U aEueqr e^r ueqlr '1eq1 sarldurr l'g €ururerl

'PeJrsep se

'lcJo r3!og =

.6 = (!X(!y)dan--- --

ul€lqo ailt (6'8) pue (8'8) tuo.r; 'fy - Z pue'!X = A"ctg/g = XEurllas'relnarlred uI'0 = lZ'Xl = h'X] r"qt 'n|;j{lX} of spleg rotra^ Zpue 'l'y se eqel a,ra y '1eql aloN 'Z pue 'r{'X sples rolf,el Jo las frela .roy

(O'e) (Z 'X|A)a n, - (Z '1,+,. Xl)d ̂h - (Z .A)d /$oX = (Z (a)a noxT

(e's)

leqt apnltuoc aa,r'91'2 ruaroaql dq 0 = dary ''a'r'uerralqey sr dzllo acurg

dilry (#) r + (an. (#) ,), = dn o(c'e'd,\

sarlr3 elnur.rog s(uelreC 'H 'ueql 'y o1 lcedsar qll^r elrl€Arrap el1 aql eq x7 7a1

.o = (,cm)p= ((,#) "^,) n = ("^, (#),),

el€q a^r 'se,rr1e,r,trep elT arll Jo uorlrugep eql urorJ 'f11eurg

'$ - d,746(1c'e/e),

6ZZ"lnruroJ s,lrad1ol4'6'9

230 8. Fenchel-Nielsen Deformations and the Weil-Petersson Metric

F ig .8 .3 .

j5 : ,S - S*

j s ( [z l= lz ] , l z leH l rs .

f,: r(Q -----* 7'(R)

given by

J ( lS , f l ) = [S * , Js o f oJ ] , [S , / ] € " ( rR ) .

This .7 is an anti-holomorphic automorphism of "(.R) which fixes the base point

lR,idl.we can easily show from the definitions of J and the weil-Petersson

metric that gvp is invariant under "7. Furthermore, we have the following:

Lemma 8.8. Denole by J* the pull-back operalor induced by J. Then

J. (dlci) - dtc;,

J.(drc) - -drcl +\ab,, ni QZ

for eaery i ( i = I , . . . ,3g - 3). Further, Qwp sat isf ies

J*(uwp) - -uwP. (8 .12)

Proof. The assertion (8.10) follows by taking the derivative of both sides of

l c i o J = l t ( c ) - l c i .Next, for'every j, we see that, though J(Cj) = Ci as points sets, the orien-

tation of ci at p is the converse of that "t J(p) for every p e T(R).II is clear

that rs, is determined modulo tcj12. Hence, we have

r c i o J - - r c i + ? 4

with a suitable integer ni, which implies (8.11):

by setting

Then we have a mapping

(8 .10)

( 8 . 1 1 )

'r - dr ; | > E! r qrr,u u,r [u]"rT ir?,{#ll.',$';':;fi""tf,':"rt.{+E-te'!1e-6to _ -

( "tg ,

'"g) dtuo - =

\0 I /

(2:g-,-'"p) FLtn+t)-\ e e )'

( (ru\ ,r- , ('"'s) ."-) dtu o-\\ s / " \ e ) " )

('"tg , bg\ dam- ,tre-oe,!*e-6to

\s I l

teql apnpuor an '(61'9) ,tq ecuag '(g - dg' ...'I = f) ! frera ro3f cto / !c.tp\-=-=1-;l-t

g \e, ou"

,o"te g *t"vQ = (t"tg\ .,elu e \e/"

ul€?qo e,rr '(II'8) pue (0I'8) ur lenp aq1Eu11e1 '1xag'g - tg ) q'[ ] I qry'{ {'f il" roJ

,rs = ffi

= (#,?) d /r,o = q' !*t-6to

eleq ea 'p'g uraroaq; o1 ,(le11o.ro3 aq1 ,tq '1srrg'71 go lutod paxg P sl

qctqa '1urod as"q eql le pereplsuoc eq o1 erc ^roleq suorlelar 11e ,acue11 .1urod

a$srq eql le "1nturoJ eq? ,lrorls ol seclgns y 'aro;eq sv 'g'g ut?ro?lJ lo loo.t4

'(Zt'g) saqdurr srqS

'(7''X)d^ro - -

(a,ylau6 _ _

(.t.t,(Xl),t)da6 - = (a,y)(ann*S)

leql epnpuoc eu'f, repun lue-Irelur sr d/116 acurg '(;61)*/ - = X*tg leql ees ein 'crqdrouroloq--rlu€ sr ll ecurs

'(I*t'X*t?)dtur|-

( A*t' x *t) d u r= (A' x) (d ilo *,C)

e a uaql '(g); uo splag

rtz

urelqoroloel luaEuel frerlrq.re aq ,4 pue y 1a1 'fgeurg

"FruroJ s,lrad1o11'g'g

Zg2 8. Fenchel-Nielsen Deformations and the Weil-Petersson Metric

l a a \aih =@w p \a/c,

, 6t", )l A n ; A 0 n x ? \

u w P \ a t c t + ; a r " , ' W * T - a r ^ )

- - a i k .

Hence, we ge t a i r=0 fo r a l l j , f t w i th 1< , t < i <3s-3 .Therefore, we have proved that

3g-3

@ w p = D d t r o - " * i A d r � i ,i = l

as desired.

Notes

This chapter follows wolpert's papers 12471and, [251]. We remark again that the

sign of the parameter t of the FN deformation is different from that in thesepapers.

For some applications of Wolpert's formula, see Appendix B.5'Several potential functions of the fundamental form uyp of the Weil-

Petersson metric have been obtained, for instance, in tomba [236], Wolf [244],wolpert [254], and Zograf and Takhtadzhyan [264], [265], [266]. See also Jost

[A-49], and Takhtadzhyan [219].

tr

1-yz-tz)

Eurddeur eql raprsuo? ea,r 'esod.rnd flql rod .dq uo fluo grJo arn?cnrls xalduroc aq1 Eurur.ro;ap ,tq ,(lrl

lpurs flluarcgns qlran) r .ralatue.redxelduoc e uo Surpuadap sacegrns uuetuerll Jo {rU} flrureg " ??nrlsuoc a \

'oO lo,{repunoq

a^l?€lar aql dC fq elouaq .(g)r_, - oO pr*,{I > lrll C ) z} = g ?eS '{Z> ltll C > r} = (n), pue 0 - (d)z ryqlaunssv .A ) d lurod uar,i8€ punore

(z'2) pooqroqqSrau al"utproo? e xrd 'ac"JJns uueruerg f.rerlrq.re ue eq U laT 'asec 1ecrd,t1 1nq aldurrs s ur uorl"rr"A rotrelul s(JeJrqcs urc1dxa Ipqs aA\ .(rhrlp

pcol (a:ou .ro) auo;o rorrelur aql ur uorleruroJep e $ uorl"rre^ Jorralur s.rasrqcs1nq 'frepunoq aql Jo uorleuroJep " sl uorlsrrel s.prsruepeH ,Buupads z(lq3nog

uorlBrJ€A rorJelul ssraJrqJs .I.v

'uorlerJel l"?rsselt l€luauspunJ go adrtl

rar{}oue 'sacey.rns uueurarg Jo uorleraueEap ssncsrp II€qs e^r ,g.V rl ,,(1eurg'flanrlcedser 'e'y pue I.y

ul (Z ?) d snua3 go d; aceds rafinurqcreJ arll Jo arnl)nr1s xalduroc eq] ecnpor]uro1 1uaun3.re (sroJlqy pue uorl€rJel Jorralul s.Jagqcs fgarrq ureldxa all

'uorlerr?A s(prerlrepeH 3o uorlezrlereueSraqlou" s€ paraptsuof, osle sr sq; .sSurdderu Jellnruq?lel ,(q pecnpur (suorleru-roJep lpurs,, Sur.raprsuoc {q ,s.ro;1qv .1 fq pacnpor?ur ,(11s.rg ser\{ U ac€JJnsuu"tuarll pesolc € Jo (U)J aceds .ra11ntuqclatr eq? Jo arnlcnrls xalduroc eq6

'([ru-V] .racuadg pu" raJrqrs prc ,[g6-y] tu"rnoC yo xrpuaddy ;c)

uollerr€A s(prsruspeH plueu€punJ pue l"clssep alotu eql Jo uorlezrl€raua3 e sePeJaprsuoc il qcrqa\ 'uorlerle,r Jorrelur s(JeJrq?s sr uorlsrr?A ltscrsselc lecrddl y'sac"Jrns uueruaru Jo suorlsrr"A Iscrss€lc se paleSrlsaAur uaeq p€q .(uorl€turoJapIl€rus,, qcns 'dl.repdod pesn aq o1 ueSaq sSurddeur l"uroJuocrsenb alogag

'U a?€JJns ua,rr3 aql Jo ((uor?tsruJoJap lletus,, a?slnurJoJ o1 fluo peeu a&r ,r(roaq1

Fcol aql .ro; '.re,ra,lro11 'U eceJrns uueruarlf uaar8 e p (U),2 aceds rallnurqcra;aql uo saleurprooc FqolE ecnporlur o1 sfeat lere^as pessnrsrp e eq aA{

SaJEJJnS uuBr,rraru uo suorlErJB^ I€rrssBIC

Y xlpuaddV

A. Classical Variations on Riemann Surfaces

on U for every €. When lel is sufficiently small, z,(Cp) is a simple closed curve(actually an ellipse in this case) in the z.-plane, which is denoted by C. and z,gives a conformal mapping of a suitable neighborhood .4. of Co (see Fig. A.1).

Now, delete Do from ft, and paste the domain D. in the z.-plane surroundedby C. .More precisely , set Vr= DrUzr(Ar) , and g lue (R- D) U,4. and I / . byidentifying z,(A,) and ,4. under the mapping 2.. Then we have a family {r?.} ofRiemann surfaces depending on the complex pa,rameter e , which is a special caseof Schiffer's interior variation. Here, note that, considering % as a subdomain offt., we can take ze as a local coordinate on V..

t ,v I".

z(A)

'1",,ffi-, it----- ' , . .rsi

z.- Plane

Fig. A.1.

When several mutually disjoint points, say ?r, ..' ,pn, a,re given on .R, wetake a coordinate neighborhood (Ui,zi) for every pj so that

zi(Pi) = o'z i ( U i ) - I t e C l l z l < 2 | , i = r , " ' , n ,A i f i U p = $ , i + k .

Set Di = z;r(12 e C I lzl < 1)).For any complex number ei with sufficientlysmall leil, consider the mapping

,(q uarrE q (U)g > 6)(!tg/f0) pue ,.lA uo Eurddcur erqdrouroloqe s! d uaq; ''rl = (r),1 ,tq I(U)A +- ful:g Eurddeur € eugaq.g_rgC ufu6r.ro aq? Jo ,,14 pooqroqq8rau lpurs f11uar?gns e xl,{ .I.y lul,e.toeqJ-loloo.t4

'ur3r.ro eql ol reau flluarcgns r fra,r,a .ro;

ro r'-i[fi-a)d 'o

\ _,o.,,,t-69,...,I - g'fq)d,!rp/!zplt )

\ /

'a o+ a;o Eurddeur l"uroJuof,rs"nb e sr ,/ reqr .^"#::rfr"rt".ffit:'i.1TJt+

las a t ' fg ;o ,(repunoq eql uo r' lz qlr^r luaprruroc pus , { g uo snonur}uoc

'{f > l14l C) r} = lguo *Cssel)Josr,'f qceg.(g-Oe, ..T -) f,fraaaro;

lZb+lz=(!z)t'll

1nd aaa '1'y uaroeql 3o goord e aar3 o;'7 .raldeqp '[Og-V] E ll

pue '[6/] reurpreg osl" eas 's^{olloJ se uorl"ruroJap leurroJuocrsenb e fq uorle-Ir€A rorrelur s(ragrq?S Eurluasa.rdar ,(q uaaoqs fpsee sr llnsar l"crsselc sllJ

'o.taz fr41ocryuapt st ld fi.r,aaa 7o|utysruoa (U)zV ur-lueutap fruo tt fr1uo puo tl salou.proo? 1oco7 ctrltLtou.tolorl lotualsfis o sea$

,t_=ol{!d} rol ['AJ +--< t |utdilout ay! lo ?sr?aul at17 ,l,aaoa.to14J

('pa47nuo y.totacuaq s, q?nln'6uu1.r,otu pau{ap Qp.tnTou ?qt yryn paddnba sr, 'g ,atag) .@),t

1" Tutod asoq ay7punorp se?ouNpron 7nco1 cttlil.toruolorl lo ruaTsfis n saat6 b = (A)l oyu.t 14 lo

['u] *------,,0uttlrlotu aqy to as.taaut eq? ueqJ .g - t lo (e_7,eC ))tA

poorl.r,oqq,tau lotus fr17uarcfins o ul , puo ,t_=u[{tay u,o$ a.rolaq sD pal?nr1,lsuo)

saeoltns uuowary lo filptuot eto ?q {'alt 7a7 :ior\ipuoc |urmoylo! ay| sa{nlos ycrynh) ld qw^

,\[{ra} sTuroil lo 1is i sTsrza er?qt uayJ 'i1uo.r7tgt, uaarf aq 't

fi.r,aaa .to! {a Io h pootl.roqq|nu o puo 'g "o ,t-;[{{a} sTurod ..urrsrp fi1lonlnug-tg IeT '(?,7) 6 snuaf to ecottns uuouLery pesop D aq A IeI .I.V uraroaql

('FZt] ll"d aas 'acuelsur .rog) 'leluaurepur-rJ sl r.uaroaq? 3uralo11o; aq1 ,,uoN.(rr, . ..,Il) = r s.ralatuered xalduroc

uo Surpuadap se"eJrns uueruarg Jo {'U} ,tFu"J e lcnr}suoc uee a/$, ,aro;eq sy

'ts)d ''T *(d)tz=tr,

'to "'-1[11-s 3 d, 'o

\ _ ,o,,, g-ft'...'I =X'!g3d ,(d)!zo)'!tor_1t'tr1 1

\ "

9t7,uorl"u?A rorreluJ s.ra$q)s '['Y

236 A. Classical Variations on Riemann Surfaces

On the other hand, by using the mean value property we obtain

t l t(21)d.21 AEj - -2rir!(pi).

- 0 , $ e A 2 ( R ) ,

should be zero.The second assertion is seen easily by linear algebra. As {pi}}o=1" in the

first assertion, we can choose a set of points such that det(g1(pi)) f 0, where

ler\?,!=1" is a base for .42(,R). tr

A.2. Period Matrices as Moduli

As stated before, the first introduction of the complex structure of To (g ) 2) was

based on investigations of period matrices. We shall show Rauch's variational

formula for the period matrices, following Ahlfors [5], and explain how to get

local coordinates of ?o by using this formula.

First, we recall some fundamental terminology. Let R be a closed Riemann

surface of genus s (> 2).Fix a set of 29 simple closed curves on ,R which induces

a canonical homologg Dcse, i.e., a canonical base of the first homology group

H1(R,Z). In this section, we use the notation {Ai,Bi}ni=, for this set (see Fig'

A .2 ) .For every [S,/] e "(ft) = Te1 we have a set {/(.4i), f(Bi)}tt of simple

closed curves on .S which induces a canonical homology base on S. We denote

this set by the same notation {,4i, Bi}oi=t.

Next, on every ,S, there exists uniquely a set {di}f-, of holomorphic Abelian

differentials, i.e., holomorphic l-forms on ,S such that

- 6 i * , j , k = 1 , " ' , 9 .

Here, writin e {, = ,lriQi)dzf on Bi, we set r!(p1) = /i (0).Now, to prove that F is biholomorphic in a neighborhood of the origin,

it suffices by the implicit function theorem that {p1}f!;3 gives a basis of thetangent space ?6(?(.R)). Since 7r("(r?)) is identified with .42(.R)- by Theorem7.5, this is equivalent to assert that any complex vector ("r, "' ,csc-s) satisfying

, \ l d 4 / d r i , p € D ir i t P l = f o , p € R - D 1 .

ll_r r,= Il,,

(8",,,,r).

Io.t'we call this {di}f=, the canonical base of lhe space of holomorphic Abelian

differentials on S with respect ro {A1,Bi}oi=t' F\rrther, we put

('pt7ua,ta$tp ctTotpanb o so paptofie-t s9 t6lg Tcnpo.ttl aqt 'any)

aff'd . tgtg I I = lt[o(rfrlp) JJ

qonbe puo 'sqstaa Tutod

esoq ?qt 7o rt uo4cer,rp eW u? rly frteaa lo frllo(tftp) aatpauap aq1 '*g ssolc

to (U)A a il fr^r,aaa.rogr (elnru.roJ lBuo.rlBlru^ s.qcnuig) .g.V uo111sodo.r4

'1urod aseq eql 1e qql e^oJd ol sa?gns 1r ,1urod es€q aql Jo uorl€lsuerl

e Ag 'il yo itlgqellueraJlp xalduoc raoqs alu ,uorllasse lsrg eql elord o;

'zalo Tutotl finaa 7o t lua.r f)u,wotil eqt soq IIp

'7, - 6 uaq74'sacopns a4fu11andfrq o7 iutpuodsa.uoc esoql ut Tilacxe Ealo Tutorl fruo 7o g- 69tlurv lorutxou eql soq II,Io Up ea4oauep eql u?!1 ,?, < 6 uayn ,taaoatoy4l

'ctyil"toutoloy ry tg *- oJ t II |utildnu eVJ .Z.V uraroaql

'ruaroaql 3uraao11o; erll Jo;oo.rd e all3 all('[gZ-V] ery pue s"{reJ pue '[g-y] orres pu? sroJIqV ,acuelsur

roJ eas 'slF?ap arour rog) .acods-t1ot1.tediln p,ary a{l sI (z/(r+r;aC )) rg ,e.ra11

'(u)-r > [/'s] '(s)z = ([/'s])z,tq paugap

og *- (a),J, n

Surddeur € urelqo am 'eaue11'elrugap a,rrlrsod q (S)Z yo 1.red {reurteur aqt t"r{t pue ,cr.rlaururr(r q G)Z1eq1 sarldurr uorl€lar por.rad lecrss"lc eql lsqt il"rsg

't=[{l7,lv} ol lcadsarrllll'A ,S lo rulotu pouail loe,ruoao? aq1 (rfz) = (S)Z x-rrpur 6 x t $ql iltsc e,1t

'6( ...'r = r'f

'z'v'ttJ

"[ =,,o

LtzInPon s" sarulel{ PoFad 'z'v

A. Classical Variations on Riemann Surfaces

proof of Theorcm A.2. From Proposition A.3 and Ha"rtogs' theorem (cf. Bers

[A-14]), we can see the first assertion.

Assume that g ) 2. Then the classical theorem of M. Noether gives that, if

,s is a non-hyperelliptic closed Riemann surface of genus g, we can find a base

of Az(s) among the set of products of two holomorphic Abelian differentials.

Hence, by the same argument as in the proof of Theorem A.1, we have the

second assertion.Finally, when g = 2, we can see directly that the set of products of holo-

morphic Abelian differentials spans .42(R) for every closed Riemann surface S

of genus two. Thus, we conclude the third assertion. n

Furthermore, Theorem A.2 implies the following:

Corollary. The complex slrttclure of To introduceil in Chapter 6 is lhe unique

one uniler the condition thal the canonical period matrix nloaes holomorphically

on To.

Remark. Besides {r1x}, Ahlfors considered integrals of holomorphic Abelian dif-

ferentials along suitable l-chains, and succeeded in introducing a system ofhole'

morphic local coordinates at every point of ?o (cf. Ahlfors [5]). This was the first

introduction of the standard complex structure of ?r.

Proof of Prvposilion /{.9. Fix a smooth Beltrami differential p € B(R). For

every complex number e with ll.pll- ( 1, let f , : R ---- E. be a quasiconformal

mapping with complex dilatation ep,. Let {|i,r}t=t be -the canonical base of

holomorphic Abelia,n differentials on .R. with respect to {Ai,Bi}J=t.Fix j arbitrarily. We set

w = ( f , ) * (01 , , ) - 0 i , o ,

where 1{]..7.@i) is the pull-back of 0i,,bV /.. Then c,r is a square integrable

closed differential on .R, and we have

k = 1 , , . ' , 9 .f t lI , = | e i , , - I 0 i , o = 0 ,

JA* JA r JAx

ffence, the period relation implies that

( r , r * ) = [ [ - u A a = 0 .J J R

Let zrbe a generic local parameter on .R.. We write

(A .1)

0i,, = ai,r(zr)dz,

with holomorphic function ai,e .Letting z be a generic local parameter on -R, we

decompose c,.r asu) = ur + @2; (A '2)

',{larrtlcadsar rzU pue IUr uo selrnc pasolr alduns Jo sles aq l="1u11+t0g,q+t6v}pue t rf{fS,(fy} pue'sacegrns uutsrueql-pasolc oml eq zgr pueig, 1a1

,1srrg'g xpuaddy eeg 'aceds rTnpou aq1 go {repunoq eql o1 Sur8rarr,uoc saouanbas;oadflolo.rd e serrr3 pue 's3urddetu

IeuroJuo?rsenb o1 enp asoq? tuor; }uareJlp fll€ll-uasse sr uorlerr€A uq; 'paleErlseaur fldaap uaaq a^€rl 'sace;.rns;o suorle,rauaSappelF)-os 'sace;rns uuetuaru Jo uorlerJe^ Ielueu€punJ pu€ Iscrsselc raqloue sv

sareJrns uueruerll Jo uorl€rauataq .g.y

tr 's^tolloJ uotlress€ eql snql

'(rlrl)o= llo'{dll . ll0'rBll . -ll'/'ll - t -TIitiEl >

l,tr ''ro'rsa[[l; Ptul I JJ I

urelqo aan '(g'y) urorg

.qtr.t + il . o't6o'tt6" [ [, =16;rfi - e),rt:t JJ

s? a)uereJrp srql ssardxa all

.zov 0"t0"[[ =(g* ,o'rr;- = (9)r{r - (a)rt:tJJ

eleq atr 'uolleler porrad eql ^q ure3e ,,t1eurg

(qv) #ffifu>l,,ll'(g'V) dq '.re1ncr1.red u1

(r.v) ._llrrrll_r; llroll llo'toll /

?eql (Z'V) uorJ ^\oqs u€? e,lr ,{lrlenbaur el3uerrl eq? pu€ (g.V) fg'zorlt - co

a^?q e^l uerlJ' zp(z)"('l). @)'I o )'lp = zO

las a^t '�lxeN'f, qcee rc1(fo'!o) = ,llroll eraq^\

(s'Y) '"llznll = ,llrrllo1 luep,rrnbe q (I'v) l"ql slrorls uorlelnduor eldurrs y

.zp(z)'('l) . ?)'t o )'!o- zo

, zp ((z)o'{o - (r), (l) . (t)'l o '' fo) = ro

6tZ setr"Jrns uu"ruerg 1o uorlerauateq .t.v

240 A. Classical Variations on Riemann Surfaces

which give a canonical homology basis on Er and ft2, respectively (see Fig. A.3).

Here, gi (i =1,2) is the genus of Ri, which we assume to be positive.

R, (g,:1) R" (s":2)

Fig.A.3.

For each j, fix a point p; € Ri, and a coordinate neighborhood, (U1,zi)

around p;' such that zi(p)= 0 and ti(Ui) - Bi = {z e C lkil < 1 }. For everv

complex e with 0 < l.l ( 1, we set

( J i , , = U i - { z e C l l r i l < l . l } ' i = 1 , 2 '

Then, identifying U1,6 and Uz,, by the mapping

Z L ' 2 2 = e ,

we obtain a Riemann surface E. of genus g = gr* 92 (see Fig. A.3). Note that

s > 2 .when 6 = 0, we take as .Ro the closed Riemann surface with a node (which

comes from the identification of p1 € ftr and Pz € Rz'Also cf' Appendix B)'

Thus we have constructed a family {8. I l.l < 1}, which we call a degeneration

to Ro with respect to (t/r, u2). on every -R., we consider the canonical base

{0;.rlsr:, of holtmorphic Abelian differentials with respect to {Ai,Bilt=r, and'

;#;;;h" canonical period matrix II(e) = (r1r(e)). Furthermore, the followingva,riational formula is known. (For the proof, see Fay [A-30], and Yamada [261].)

'[I9Al spelus^ uI Ifre1o.ro3 ees 'Erelap aql otut a^lep lou op ain 'rarraiuog .r ol lcadse.r qtur turelrapro ?srg eql pue '0a pue X Jo senlsA eq1 dllrcqdxe urlrop alrr^\ ppoc eA\

'lur,lsuoo elqDpns o s, b puo'.toycaa

f)uorsu?utrp-(t -n) o c! X tg uo tt"tTout pouad ID)tuouD? ayq st, oy ,ata17

.(o*l,l) e)o+foe+'jq+ #l =t,lzL ^t -u)

.g.Y uraroaql

([196] epeu"A pue '[Og-V] feg ya) ur\ou{ sr €lnturoJ l€uorterrel3urno11o; eql ueql '(r)7 secrrleur por.rad lerruouec aql auuep erlr ,7.V .3rg ur seug pw ty Sursooqg 'f snueS;o e?eJrns uueruarll pasolc € sl ,gr uaql ,g

f t;1

.?.V'EIJ

'(zn'rn) o1 lcadsa.r qll/'^ ry oluoNllreu?fiap elpt osls e!$, q?!q,!\'{I > lrl | ,U}dlturey e lcnrlsuoc uec e^ 'a.ro;aq s-e fem etu€s eq? ur 'uaq; 'Q = zn U r, leqtPu€'f qcee roJ {r > | tzll c ) z} = (frilt, pue 0 - (!d)!z wqtqcns 161 = .r;fd puno.re (!r'12) spooqroqq3reu alsurprooc pue 'g uo zd pue ld slulod 1?urlsrporal xld 'eseq ,(Eolouroq Iecruousr e secnpur qclr{rt U uo se^.rnc pesolc eldturs;olas e eq FI{\A'fy} pue '(t <) t - f snuaE;o aceJrns uuerueql pesolc e eq Ur lel'flesreard erotrq 'feal Jelrrurs e ul zU = IU leqt aspJ erll l€erl uet a,u, ,1xa11

'lg uo qo4ua.tafitp uDrIeqV cryd.totuoloy to asoqf)cruoun? aq1 st r=,ro{t'!61 a.teyn '(!6' ...'I = {) l2 uo lzp(lz)r'to - t'!6 yt!,n

((o)'o'rr' .'.'(o)t'ro) = rx

puo 'fg uo eulout pot.tad lorruouo? aq1s9 ly'2,'T= t qcna.tol'ate11

'(o-lrl) (.rlo+f- 9 tXzX'1 |zrr nl

'u Lzxtx, o 'l't"z-

L ; '|rl =(')r'7'V rrraroaql

rt7,saf,"Jrns uu"urarlf ;o uorleraua8eq't'V

242 A. Classical Variations on Riemann Surfaces

In the described degeneration to ,8o, if we take aspecial (Ut,Uz), and restricte to the set {qtz | 0 < t < 1}, where 4 is a constant with lql = 1, then we obtainthe so-called Schiffer-Spencer's aariation (cf. Schiffer and Spencer [A-94]).

A variation corresponding to the second case is called the uariation byattaching a handle, which we shall explain more closely.

Let .R be as in the second case. Fix two distinct points p1 and p2 on .R. Fixalso a point po € R- {pt,pr}. Then there exists uniquely a harmonic functionG(p) on R- {pt ,pe} such that :

(i) G(ps) = [,(ii) G(e) -log(I/lz{p)l) it extended to a harmonic function in a neighborhood

o fp1 , and(iii) G(p) - loglz2@)l is extended to a harmonic function in a neighborhood of

P2,

where zi is a local coordinate with 4(p1) = 0 at pi for each 1. We may call this

G(p) the Green function on R normalized at ps with positive pole at pl and with

negative pole at p2.Now, for a sufficiently small positive t0, we put

andUz = {pe n I G@) < logts} u {pz}.

Then [/1 and U2 are both simply connected domains. Fix such a t0' and take

these domains as U1 atd Uz in the second case. As a local coordinate, say (j ' on

Ui, we choose one such that

R " ( i ( p ) = e x p ( ( - 1 ) i G ( p ) ) , j = r , 2 .

Fix a complex constant 4 with hl = 1, and consider e = qt2 for every t with

0 < t < ts. In the sarne way as before, we can construct a family {&,r, | 0 <

l ato\. This is Schifer-Spencer's uariation bg attaching a handle. We know the

following classical Schiffer-Spencer's aarialional forrnula. (For the proof, see for

instance Schiffer and Spencer [A-94].)

Theorem A.6. Let Q1 and q2 be arbitrary d,istinct points on R- \po,pr,pzl,-Let g1(p) be the Grcen funclion on R tz normalized, at ps with positiae pole at q1

and with negatiue pole at q2. Then

1ct(p) - go(p) =

#*"tol(c(c') - G(qr)) + l2Re (qc) + o(tz) (t * o).

Hew, c is a constant independ,enl of t, and lhe conaergence is locally uniforrn on

R- {P r ,P " ,Qr ,9z } .

u, = {oe a

I c6y' toe!

} r ,o,r,

'g xpueddy osp ees 'saceds llnpour peg-rlceduroc aq1 ;o fpnls aql ur Iool plueuepunJ ? sr sa?€Jrns go uorleraua3aq

.lvzzl pu" [966] rqcnSrueJ eas .ec€Jrns uuelueru fre.rlrq.re ueroJ prle^ IIIIs sl elnuroJ s,racuadg-ragrqog 'rarroaront '[qUZ] pue IVZZI rqcn3rueaeas 'sacrJleur porrad l"f,ruouef, aql roJ araq pe^rJap esoql ol relnuls selntu-roJ leuorlerJel urelqo u€f, pu€ 'suorle.raua3ap pue suoll€turoJep leurro;uoctsenbSurleureEleure dq suorlerrel raprsuoc u€) e/r\ 'sace;tns uueruelg frcr1tq.re rog

'([gZA] rqcn8ruea pue [IfI] e{e}qo ';c) ,,a3.re1 f11uercgns,,

q ar l€ql erunsss plnoqs a,u '(ar)J;o lurod aseq aql Jo PooqroqqErau e sra^oc

{A -U uo leuroJuor q.f | (U)-r > [/'S]]

leql epnl)uoc ol repro ur 'ueqA 'U Jo lesqns e eq g 1a1

'eldtuexe

roJ 'ploq fgressaceu lou seop I'V rueroaql se uotlJesse u€ qrns 'leuorsuatutp

flalrugur sr U a?€Jrns uu€tuerll uado uefo (U),2 eceds rallnuqrletr eql uaq1\'[696] rqcn3etue pue 'lt1Zl'lgZel rqcnSrue;'[eOt] '[fOt] eqlqs'FtI] Iue]IetrAtr

alrc osp aal'sace;.rns uusruorg le.leuaEJo as€c eql ur suorlerrel roJ sy'[661] ruel-r"tr{ pue l{ounsny errr€lsur roJ aes '(p)g eceds Jellnurqclal aql uo i(lerrqdlour-oloq eloru ..secrtrleru porred Iecruouec,, eseqJ 'sl€rlueragrp u€ITeqv ctqd.rotuoloq

Jo ror^er.leq drepunoq aql uo uorlrpuo? elqelrns e Sursodurr .{q ,,xtr1eur pouad

Iecruou€c,, eql Jeplsuoc uec eA{ 'g ace;.rns uu"ruerp frerltqre u" roJ ualg'rl

le.rauaE " roJ sploq llrls llnser eq1 'ralano11 'rl lerlua.ragtp

rurerlleg qloous e fpo reprsuof, a,r,r '{lrcqduls Jo a{ss aq} roJ 'g'y uotltsodord uI

seloN

saloN wZ

Appendix B

Compactif ication of the Moduli Space

Following Bers [32], [33], [34], and [40], we shall construct a compactificationof the moduli space Mo of closed Riema^nn surfaces of genus g by adjoining toMn the set of biholomorphic equivalence classes of closed Riemann surfaces ofgenus g with nodes.

8B.1 Compactiffcation of M1

As an example, we construct a compactification M1 of the moduli space M1 oftori.

As was seen in the remark in $2.1 of Chapter l, My is identified with thecomplex plane C, and every point in M1 is represented by the biholomorphicequivalence class [.91] of the torus ,S1 defined by the algebraic equation urz =z(z - l)(z -,\) for a complex number ) with ^ + 0,1. A compactif ication M1 ofM1 is the Riemann sphere e = CU{-}. I lere the point oo in e correspondsto an algebraic curve given by the equation w2 = z(z -l)(t -,\) for ) = 0, 1, or@ .

degeneration-------)( l - 0 )

Fig. B.1.

For example, taking I - 0, we see that the algebraic curve ,9o given by theequation w2 = z2(z - 1) is the one which has a sole singular point at po = (0,0).

) - n

'(Z'g'31,{ aas) g uo epou qrea Sutuedo fq sapou lnoqltm d'snuaS;o og acegrns uueuralg pasolc e 1aB e.n l"ql sueau (rrr) uorypuoJ'qrou.tey

r=!"7-o '{-I+-*''.r.

uaql '.{laarlradsar '9, yo slred pue sapou Jo sraqunu er€ { pu" u/ JI (II)'.i? ec€Jrns Suuarr.oc IesJeAIun aqt seq fg, 's1

1eq1i0 < lu + Z - !62 segsrles pue '16 snua8 go e?€Jrns uueruelu pesol? e uro{slurod l)ullslp fu Sutaoruar,{q paurelqo sI qclq.t\ eceJrns uueuratg e ''a'r'(!u'!0) ed.{1 e1rug.{11ecr1,{leueJo e)eJrns uu"urarg € sl Ugo fg, 1.red dleag (u)

(g 1o 1.tod e pellec sr sepou slr II€ snurur g;o luauodtnocpalf,euuoc V

'ellug sl U Jo sepou Jo u, raqunu aql 'lf,edruoc sr U ecurs 'Ur

lo epou € pell€r sr d'asec ratt€l aq? uI)'{I > ltrl P,tn t > lrzl I C x C) (zz't7) ) ras a{t to {I > lrl I c f z} {slp }Iun eqt o1 crqdrouoauoq er€sluetuale esoqn\ spooq.roqq3reu l€?ueuepunJ Jo ure1s.{s s seq U 3 d ,{re,tg (t)

:suorlrpuo, aa.rq1 Sur,r,rolloJ aql segsrl€s A I sepou q?!n 6 snua| toacn!.tns uuDurexv pasop p, palpt $ gr eceds JropsneH pelf,euuot lceduroc y

'(Z 7) 0 snua8 ;o sa?eJrns

uueuttrerg pesol) Jo 6W aceds qnpour eqt Jo uorlecgrlceduroc e ltnrlsuoc eA\

(z6 "roy 'W Jo uorlergrlceduro3 g'g

'og se papre3ar st uotlecgtlceduroc lutod-auo s1tpue '{

O } - C o1 luelearnba fqecrqd.rouroloqrq sl -J/C e)eJrns uueruelg eql'g q3norql oo - r se rl

;o uorleraua3ap aqt ,(g paurelqo sl rcJ lerll ees a.!r'(Z'Z)lSa roJ J urerrrop l"lueur€punJ e se 6'I

'3lJ uI €ar€ pap€qs eq1 3ut1e;'H ) r aruos roJ I + z - (t)'g't * z = (z)',L suorlelsuerl o.rl r(q pale.reua3dnor? acrllel aql eq | 1a1 :ssacord Surarolloy eql fq peurclqo sr 6,_;r dnor3 srq;'I + z - (z)9 uorlelsuert eql .,{q pele.raue8 (C)l"V yo dnor3qns e sl -J eraq^r'*J/J eceJrns uu"uraru eql ol spuodsauoe C f oo lurod aql ueqJ,'I .ra1deq3

Jo I'Z$ ul s (Z'dlsaln eceds luarlonb ei11 qtlm I,ZV ,ty11uap1 e.tr '1xetr1'0 * y se od epou e olul Y5'uo

(y)Iy a^rnc pasol) eJo uotlerauaSep aqt,tq paurelqo sr 1r pu"'I'g'3lJ a rl s)looloS: l€r{l ees a^r 'uorlerr.rasqo srql uord 'epou to 7ur,od alqnop fuoutplo ue pellelsl ,9Jo odlurodreln3urse qcns'peyluapr are 0

- ,Mpue 0= Z slutodorr,rl

areqr'n'{ t > lUl I C > U} p"" { t > lZl I C > Z} rr{"lp o^rl Jo uolun eqt s€papreSar sl o5l slql'od;o pooqroqq3reu e ur'snq;'?uelsuof, e,rrlrsod " $., ereq^\

'{t > lul't > lzl'o -- tt4z I c x c > (u'z)}

,(q paluasardar sr og elrnc cleJqa3p aq1 'I - zfz + n = .1Apue I - zfz -m = Z uorlsruroJsuerl eleurprooc eq?.,(q'od;o pooq.roqq3rau e u1

=)

9VZ614J yo uoqetyrleeduro3 'g'g

246 B. Compactification of the Moduli Space

Fig. B.2.

A homeomorphism f : R --- S between Riemann surfaces with nodes is saidto be biholomorphic if / induces a biholomorphic mapping of Ri to a part of^9 for every part .Ri of .R. If there exists a biholomorphic mapping of .R to ^9,then rR and ,S are said to be biholomorphically equiaalent. We denote by [E] thebiholomorphic equivalence class of a closed Riemann surface r? with nodes. Asa compaclification Mo of Mo with g > 2, we take the union of Mo and the setof all biholomorphic equivalence classes of closed Riemann surfaces of genus gwith at least one node.

Now, we define a topology on Mo by using the Fenchel-Nielsen coordinates asfollows: let r? be a closed Riemann surface of genus g with rn nodes and ,b parts.As stated in the preceding remark, we can take a closed Riemann surface .R, ofgenus g without nodes and a system of decomposing curves L = { Cr, . . ., C"o _"}on -R, so that rt is obtained from fto by degenerating each element of a subset

{ C i r , . . . , C j ^ } o f , C i n t o a p o i n t . D e n o t e b y ( t . , 0 ) = ( 4 , . . . , L s s _ 3 , 0 1 , . . . , | s c _ s )the Fenchel-Nielsen coordinates on the Teichmiiller space ?, of genus g associatedwith .C (see $2.1 of Chapter 3). From the proof of Theorem 3.10, for any point(1,0) e (n+;sr-s x R3e-3, we can construct a closed Riemann surface R2,s ofgenus g such that ,Ra,e induces a point in ?o whose Fenchel-Nielsen coordinatesare (1,0) .

Here, we admit the case where some /r', ,. .. , l j^ in { /1 , . . . , lsg-z } vanish. Inthis case, we get a closed Riemann surface R2,9 of genus g with n nodes by a con-struction similar to that in the proof of Theorem 3.10. (We consider that each ele-men tC1r , . . . ,C j . i n ,Cdegene ra tes in toapo in tonR lp . ) Then .Rhas theFenche l -N ie l sen coo rd ina tes ( l (R ) ,0 (R) ) = (4 (R) , . . . , 12c -s (R) ,0 t (R ) , . . . , dsc - r ( f t ) )w i t h / i , ( f t ) = . - . = l j . ( R ) = 0 . F o r a n y ( . , 6 ) - ( e r , . : . , € B s - 2 , 6 1 , . . . , d e c - s )with positive €i and 6i, the (e , 6)-neighborhood of [^R] in M, is given by the set ofall biholomorphic equivalence class [.R1,e] of closed Riemann surfaces r?z,a withor without nodes satisfying the following two conditions:

( i ) W i - l i @ ) l ( e i f o r a l l j - 1 , . . . , 3 s - 3 ,

(i i) l9j - 0 j(R)l ( 6i for all j with 4@) + 0.

node

'.(a)pow fql! alouap pue (g)41o dno.t0 uorTonuotsuo.tT rDlnpout Jellnun!?rry aq1 dno.r8 srql

IIetr e1ydno.r3 e uroJ 'A - A : rn srusrqd.rouoeuroq 3ur,r.rasa.rd-uorleluarro 1p ,fqpecnpur (i[nl] sEurddeur eql'[U - S:looll ot [,U - g:/] spuas qrlq/" (U)d* (U)A:-[o/] Surddeu € sa?npul U * ,A:ot uotl€ruroJep 3uo.r1s,,(.rarrg

'a Jo (a)Jeceds rallnurqclaf eql qlra peyluepl q (U)C 'sapou ou seq U JI

'g' t- S I tsuorleuroJap Euolls Jo [U - g:,f] sasselc acualenrnbe IJe Jo las aqt sl U Jo(g)g acods uotTounotap 6uo.t7s aq; 'flairrlaadsar ,f1r1uapl aql ol pu€ Surddeurcrqdrouroloqlq s ol crdolouroq suuqdrouroauoq alrlce[q "* orl pue q ereq^d

et'a*zS

f^ "vl l'/

u*v ts

urer3erp elrlelmuruot e sr eJer{} l\ ryel-oatnba eq ol pres ere U F zg :zt pue U .- r.9 :I/ suorleur.rogep 3uor1s om;

'f ualr3 e ro; f snuaS;o sepou qlr^{ saceJrns uueurarg pesolc leurur.rel fueurflaltug fluo 's8urddeur crqd.rouroloqlq of dn 'ale ereql leql eloN 'sepou g - dg,,(1aueu 'sepou

Jo raqunu alqrssod 1se3.re1 aql seq ll lr lvuttuel sr p, fes e11

'rusrqdrouoeuoq Surarasard-uorleluarro ue sr UrJo sepou 11e yo sa3etur esrelur or{l Jo }ueuelduroc eqt of /;o uorlcrrlsa.r aqa (rrr)

'sepou 11e Surpro,re g

uo alrn) pesol? aldurs e ro S'Jo apou € sl U Jo apou e;o a3eur esrelur eql'Ur

Jo epou € st S'Jo apou " 3o a3erur aq;

:suorlrpuor aa.rq1 Sura'rolloJ aqlsagsrl"s llJl U lo uotyout.totap fuatTs " pallec sl U * g :/ uorlcatrns snonurluoc

V'(Z ?) f snue3Jo sapou qlllr sec€Jrns uu"ruerlr pasolt o^rl eq,g pu€ A p"l'sePou qlrAr sal€JJns uuel'ueru ,{q pacelde.r er€ sereJrns uu€tuaru pesol)

qf,rqa ur aceds .ra11nurq?latr Jo uorlezrlereuaS e 'sapou qlr^r ec€Jrns uu€tuarg ego aceds uorleruJoJep tuorls eql lcnrlsuoc e,n'1'g tuaroeql errord o1 Jepro uI

sacedg uorlBruroJeq tuo.rls g'61

'uorlras lxau oql ur paureldxa $ ueroaql slqt Jo;oold e Jo eulllno uy

'acods $.topsnog

Tcoiltuoc o st'paquasep so pe?ctunsuoc '(77 6) 6W acods aqa .I.g uraroaqtr,

'tuaroeql

Surr'ro11o; eql e^eq arrr /!!oN ''W uo fEolodolSropsntsH € sa?npur qcrqr*'t1,g ur

[g] yo spooqroqqErau leluaurepun; ;o ruelsfs e a,rp spooqroqqSreu-(g'r) aiaql

(rr)(r)

Ltzsaredg uorl"urrolaq tuorlg'g'g

248 B. Compactification of the Moduli Space

We define a Hausdorff topology on 2(R) as follows. Let S be a closed Rie-

mann surface with nodes, and C be a closed curve on a part S; of S. We set

. tslCl = igl ts,(C'),

where C' runs over all closed curves on S; freely homotopic to C, and lsr(C')

is the length of Ct with respect to the hyperbolic metric on ^9i. We also set

IIIP] = 0 if P is a node of S.Let C = {Cr, . . . ,Cr} be a f in i te set of c losed curves on par ts of S and e a

positive number. A strong deformation h: S' - S is said to be (C, e)-smallif i t

satisfies

( i ) l ts , lh-L(C1) l - ts lCl l l<e for 7 - r , . . . , r ,

(ii) lls,[h-1(q)]l < E for all nodes q of S.

We say that a set [/ in 2(r?) is open, if for every [/: ^9 * /?] € [/, there exists

a finite set C of closed curves on parts of ^9, and a positive number e such that'

whenever h'. S' -,9 is (C,e)-small, the point lf oh: St'.- R] € 2(,R) belongs to

[/ (compare with $3 of Chapter 3).There is a canonical projection np: D(R) - Ms which sends [/: S * R] to

[S]. It is seen that the canonical projection I/6 is a continuous open mapping.

Let us introduce the Fenchel-Nielsen coordinates on a strong deformation

space.First, assume that rR is a terminal Riemann surface of genus g with nodes

{ei}10=1t. Take an arbitrary point [/:,s * n] € 2(l?). It f-t@i) is a simple

closei curve on a part ,91 of S, then we can choose a unique simple closed geodesic

Li on 516 which is freely homotopic to f-t@i). lf f-t@i) is a node, then we

p,rt f, i = f-r(Pi).In this way, we have a system L = {fi}1n=1" consisting of

all nodes of S and some simple closed geodesics on parts of S.

lf f -t@i) is not a node, we set

, j = * r s * ( L ) e ; e i ,

(B .1 )

where di is the twisting parameter with respect to tri such that 0 1 01 < 2t

(cf. $2 of chapter 3), and ts*(Li) is the length of the geodesic.Li measured by

the hyperbolic metric on S1. If f-r@i) is anode, put zi - 0.

It is shown that the numbers (rr, ' . . , zzs-s) depend only on the equivalence

class [/: S * r?] and the mapping of D(R) to C3g-s sending [/: S * R] to

(rr,...,zzc-z) is a homeomorphism, which is called the Fenchel-Nielsen coordi-

nales oni@).It is also proved that for every strong deformation fs: Rt + R,

the induced mapping lfsl-:D(Rt) -D(R) is a universal covering map onto its

image, the image being the set of those [/: ^9 * -R] for which f-'@i) is not a

node whenever /o 1(p1) is not a node. F\nthermore, D(R') is homeomorphic to

C3s-2.Next, the Fenchel-Nielsen coordinales on the strong deformation space 2(R')

of an arbitrary closed Riemann surface ,? of genus g with nodes are defined as

pa)npureqrpue(a)auoaln?f, nrrs'",0*'ilT',Tfi ;T,':T"r:".1r:t'#rXTiit;: 'orqd.rotuoloq sl (("U)A)-[oI)U n ol uor]crr]ser stl Jl ctrld.tou.roloy pa11ec sr (g)gur 72 les uado ue uo uorlcunJ snonu?uoc e ''e'r 'eln1ln.r1s pa3urr e seq (g)4 ueq;'('U)A

Jo ernlrnrls xalduoc eql tuoq pecnpur arnlf,nrls xelduroc Iernleu e s€q((A)A).10t] 'ecueg 'deru Eur.rerroc l€sralrun * q ((,U)C).[0/] - (A)At.[ol)'rarroa.rotr11 'esuep eraq.nou rt (('U)C)-[ot] - @)A pue (lI)@ ur ureruop € sr(A)A ,- ("A)A: *[o/] Eurddeu pacnpur eq] Jo (("A)A).loll a3eurr eq] ueqJ'A - oA : 0;| uorleu.ro;ap 3uor1s e e{€I 'arn}rnJ?s xelduroc e seq (og)4 acuaqpue '('U)J aceds .ra11nuq)lel aql qtr^r pegrtuapl q ("U)@ 'uorlces snorrlard aq1uI palels sV 'sepou

lnoqlr^r ro r1lrrlr snuaS aures eql Jo e)€Jrns uuetuarll pasolc.req1o fue eq U lel pue 'sapou

lnoqlr/( 6 snua3 Jo eceJrns uueruarg pesolo e eqoA p.l '3urmo11o; aq1 sr ty,g go ern?rnrls xalduoc eql Jo uorltnporlur lsrg aql'sfeiu, o.tr1 uI 6W

Jo "rt1rnr1. xalduroo eql ecnporlur ol ^roq,tgar.lq aqrrf,sap aM

otr_ll Jo ernlrnrls xalduo3 p.g

'lceduoc ,\ 6try WqIapnlf,uo) e^\ snqtr '6141 rerot uW * (a)A; rtfi suorlcafo.rd lecruouec aql repun"N'"' 'I;l7

Jo sa8erur eql l"qt qrns (fy)4 3 fy slas lceduroc erc araql leqlsatldrur Z'g "rutuaT 'saleurprooc ueslarNlaq)uag Sursn 'ueqa 'd snua3 3o sepouqll/tr ?

' " ' ' IU sa?€Jrns uuetueru leurturel lualearnbe-uou II€ e4ef ino11'([6p] stag '3e) rasng .,tq ua,rr3 sI eurural srql ;o yoord

luaraJrp V'[0gI] r{qelery pue'[81] €rlr€snrrq i86'd'II .ra1deq3 ur g euural'[1-v] "Uo{lqv

'eldurexe ro; (aag 'Durutal ronoc eql Sursn ,(q pe.rlo.rd sr eruural srqtr

'E - ft'" ''I - !17o .rol 7 j (13)7 |ugfitstlos saarw |uzsodu.rocap to {t-0tg'"''rCl = jula4sfis o sorl 6 snuaf lo aco!.r,ns uuDurezy pasop fuaaa pql qcns 6 uo fi1uo|urpuad,ap .J luolsuo? aatTtsod D s?sme a.taq7'676 fi.to.t7tq.tD ro4 ,Z.B BruruaT

'euurtuel 8urrr,lo11o; aq? peeu eM'6W;o ssaulredruor aql /t\oqs ol rapro uI'usrqdrouoeuoq e

sr (e-re,7,'"''rrrl) ol [rU * g:q] Surpues e_oeC ol (,A)Ago Surddeu eq] ?tsq]u^,roqs q lI

'(t .ra1deq3 ur 6'g €urue1 'gr) ,_rgC o1 crqd.rouroeruoq sr (,g')4 acurs

'(,A)A uo q)uerq snonurluof, panpr'-a13urs € serl {z 3ol uorlcunJ aql }eq} saqdurruraroaql durorpouotu eql uoqt 'gyo apou e lou $ (ld):l;r'ara11 '(1'g) ,tq uear3

[tt - S:ttoot] Jo se]eurproor uaslerN-leq)uad eql ar€ (t-68,2'"''rz) eret{.n

seleurProof,

6VZ

',U Jo "po.t e lou fl (fd)r-o/ ft lzSol= lm'g;o apou e sr (fa)r_o;"r, lz - lm

'tq uartr3 ale (e -fe61 ' ' ' ' 'rm)

uaslarN-laqrueJ sll'(U)A > [,U * g:r1j lurod {rara ro;:s,llolloJ

6.;ig 3:o arnlcnrlg xalduro3 'p'g

250 B. Compactification of the Moduli Space

Using the Klein-Maskit combination theorems (Maskit [A-71]), we have the

following theorem (cf. Bers [40]).

Theorem 8.3. D(R) is a comltler manifold and is rvalized as a bounded domain

in C3c-3.

Now, we get the following two results.

Theorem 8.4. The Teichmiiller rnoilular transformation group Mod(R)* is a

discrele subgroup of the analgtic automorphism group "f D(R). Moreouer, the

subroup Modo(R)* induced by the biholomorphic mappings of R onto itself is

finite and is the stabil izer of f id: R- Rl in Mod(R)..

Theorem 8.5. Therv erists a neighborhood N of l id: -R- R] inD(R), inuari '

ant uniler Modo(R)*, such that the quotient space Nf Mod"(R)- is homeomor'

phic to a neighborhood of lR) in Mo.

By H. Ca.rtan's theorem, the quotient space NlMod"(R)* has a normal com-

plex space structure. Thus M, becomes a normal complex space, and it leads to

the following, the main theorem in this Appendix.

Theorem B. 6. The compaclification Mn of the moduli space of closed Riemann

surfaces of genus C e_ D has a normal compler space straclure of dimension

3 g - 3 .

The second introduction of the complex structure of Mo is the following.

For a given closed Riemann surface R with nodes of genus g, considering both

quasiconformal deformations and degenerations, we construct Riemann surfaces

which represent a neighborhood of [n] in Mo us follows.

Assume that r? has rn nodes p1, . . . ,Pm and & par ts f t r , . . . , -R1 such that

each.Ri is of type (Si,ni). Then it follows that

D n i = 2 ^ , D o r * m - k = 9 .j = l i = L

For each node po on rR, suppose that pa corresponds to a point obtained by

identifying a point oo in E* -.Ro, with a point 6o in E* - fto, for some d1 and

o2 in { I,. . . ,k }. Here, { denotes the natural compactif ication of Ror. Take a

local coordinate neighborhood (Uj, zo) at oo on E[such that zo(ao) = 0 and

z.(u:) = 4, the unit disk. similarly, choose a local coordinate neighborhood

(U3,J") at 6o on R', such that too(bo) = 0 and ."(UZ) = 4. Further, take a

relatively compact open set Vi in Ri for every i = l' .. . , /c so that I/ = l)!=rV

meets nei ther [ { nor U2" for a l l a = 1, . . . , f f i .

Set t ing N = 39 -3+rn, we can f ind Bel t rami d i f ferent ia ls P ' t - t . . . ,p7y on

R' = R- {pr,...,pm} such lhat pi vanishes identically outside I/ for all j -

- ta = "':a

) d\ - o's'an

) d| - o's'nn

1nd a6'u.L' "'

I = 12II€ roJ I > l"rl leqt qrns uc ur lurod e aq(uo' "''ro) - D !3-I

'8'g'ttJ

(".rnn)n (""u f ) '{ l'"1 t l(d)''"*ll ''zn'{ l""l 5 l(d)"'"rll "'?tl

\ sapou

x"uoAuotleurro;ap cb

€-lsrd f

uo!l€f,grluaPr

'g uo Dd epou eql o1 Eurpuodsa.rroc sgr uo epou eql t'Dd

fq elouag 's1red I prre sapou u, qlr^t f snua3 ;o

sgr ecey.rns uueuerg pasolc euptqo at'l.'utr"'t1 = rc q?"a JoJ ("9)'/ qll^ ("D)'/ Eur,t;rluapr (8'{U("''''IU

ruoq '8'zDU uo ("9)'/ yo (r-toorn'7n) = ('"n'?n) pooqroqqSreu alsurprooc €e^eq a/rd, 'flrelnurg 's'rDU uo ("a)t Jo pooqroqq3rau aleurp.rooc e sr (tjto"z'"n)- (''or'?n) trqt ees e,lr'l.nlu'o leuroJuoc s "/ ecurg'"'fU ol fUSo Surddetu

I€ruroJuof,rsenb e o1 spuelxe lA ol'[ Jo uorlcrrlsa.r aq1 pue'1!u-.nl) adfl elrugf11ecr1f1eue Jo ec€Jrns uuetuarg " q "/ .repun {g, lred e lo (!A)"t - ''!A a3euraql '( {U)J t=jU ul lpl' ,Aho pooqroqqSreu e oluo 6'go Eurddetu orqd.rouoloqrq

€ sl ["/'iU] ot s Surpues (ig)t t=j[ of,q g;o Surddeur aql t€q] erunss? feur e16

I=f'lr1 lsf : c71

n, rua,saoc rur*rlrag qlr.ryr

7A *,A : J Surdderu FruroJuorrssnb e slsrxa e.raq1'g ) (t"' "' .I") - s fuero;'uaqa 'n,C ul ur3r.ro aq1;o pooqroqq3rau uado lpurs ,flluaralgns € aq O ?eT

'(y xrpuaddv ul I'V ureroeql

:.:c) W?',U] ?ulod eseq aq? w (!A)l saceds rellnuqtlal f" (rU),f t=jLJ "rnotlcnpord aq1 ;o eceds luaEuel aql Jo srs€q e ecnpur ,taq1 1eq1 q?ns pue rN' ( ' ' ' (I

19Zo7g;o arnlcnrlg xaldurog 'p'g

252 B. Compactification of the Moduli Space

Identify any two points c and b in .Rl,o if o and D a.re contained in Uj,,,o

and(J!.r.o, respectively, for some a, and if they satisfy za,swa,s - ao. By this

identifiiation, we obtain a closed Riemann surface rR",o of genus 9 with n nodes,

where rn - nis the number of o's with oo I 0 (see Fig. 8.3).

Now, we set

6 = { ( s , a ) € D x C ^ | l o i l < 1 , i = L , . . . , m } ,

D = { [ f t " , o ] € f u o l G , Q e b ] .

Then, 2 is a neighborhood of [R) in Mn' However, the mapping of 6 onto D

sending (s,a) to [.Rr,"] is not always injective. By changing D suitably, we may

assume that the biholomorphic automorphism group Aut(R) of .R induces a finite

group G, consisting of analytic automorphisnT "jD, sl'ch that the quotient space

DlCts homeomorphic to aneighborhood of [n] in Mn . For details, we refer to

Bers [33], XIII in $7, and Masur [143], $2, and Wolpert [249]' $4. As before, H.

cartan's theorem implies that the quotient space Df G has a normal complex

space structure. Thus, I4, becom"r a normal^complex space of dimension 3g - 3.

Note that this complex space structure on M, is equivalent to the one given in

the first introduction.

8.5 Weil-Petersson K6.hler Form on the Moduli Space

since the weil-Petersson Kihler form wylp on the Teichmiiller space ?(.R) ofgenus g (= 2) is invariant under the action of the Teichmiiller modular group

Mod(R), it is regarded as a form on the moduli space Mo. This form is denoted

by the same notation^uwp. we are interested in the behavior of u.,ryp near the

boundary of Mo in Mo.

From the construction of Mo and Wolpert's formula (Theorem 8.6), uvvp

extends smoothly to the boundary with respect to the coordinates (t,r) =

( l r , . . . , l s g - s , T L , . . . , r s g - s ) , w h e r e r y = t i 0 1 f 2 t f o r X - 1 , - . . , 3 9 - 3 a n d

(t,0) -- (h,. . . ,tss-2,01, . . .,f lsg-t) are the Fenchel-Nielsen coordinates asso-

ciated with a system of decomposing curves on ft. In particular, M o h* a finite

volume with respect to the Weil-Petersson metric.

On the other hand, the boundary behavior of uw p with respect to the coordi-

nates (s, o) given in the previous section is studied by Masur [143], and wolpert

[251] .' Next, curyp induces a cohomology class [c.,szp] on Mo such that lusr p)lt2 is a

rational class (see wolpert t249]). Thus multiplying [c..,szp]lo'by some integer,

we get a line bundle over rt[o. wolp"rr l252l proved that this line bundle is

positive, and consequently Mo is embedded in a complex projective space. Hence,

we have the following result, which was first proved by Knudsen and Mumford

[117] by using algebraic geometry.

Theorem 8.7. The compactification Mo of the moduli space Ms of genus g

(] 2) ts a projectiae algebraic uariety.

'[221] punaq pu" urqeg pue'[991] uosleg'[191] uralsqtog pu€ unrgerl'[09] tlqtg pu" auerC'[96] luedrgpu" roleqrl€g aas 'aldurexe roJ :parpnls uaaq aA"rI se?eJJns uu€tuerg .radns ;osaceds rlnpou pue saceds reflnurq]lel (/tJoeql Eurrls radns qlr^r uorlreuuoc uI'[296] tradtom pu€ '[gzzj rqcn3ruea '[oo] pq!"n uI palPnls are eseql 'aceds

rTnpou aql;o f.repunoq aql reau suorlrunJ ueerg eql pus uolltunJ elez Sreqlageql Jo ror^€qeq aqt ^rou{ o1 fressacau q q erer{tr, '[911] ,ro1ef1o4 pue '[291]

uosleN '[92] ueparrg '[91] uoqap pu€ aun€C-zers^ly '[9I] ,ntnr'1y '[att-V] .,"Lo1 .rap.r aru 'crdo1 Surlsaralur qql rod 'a1or luelrodtul ue feld saceds qnpoupue saceds rellnuqcrel ;o froaql aql 'scrsfqd alcrlred ;o f.roeql 3ur.r1s u1

'[qOZ] pt" [tOA] "tolqS pue '[09I] as€tntr l'[16] elpr'neaS '[ft] o[ereqrY 'aldtnexa to3: 'aas lcafqns $ql rod 'sse.r3o.rd 1ear3e epetu uaeq s€q e.raq1 'sarlau€A uerqocsf uo uralqold s(Iqoc"f uo 'f11uacag

'[qqa] '[oqz] 1.rad1o7y1 pu€ '[9ZI] rauua4 '[gg1]'[691] proyrunry '[UtI] u€lqc"le€I I '[96] p.rolurnntr pue slrr€H '[qO] '[fO] slrrsg'[96] rar3eg prr" rarlag '[ZO] '[tO] '[66] re.reg '[99] 1.rad1ol\ pue sSurpprC '[gg]

srrr€H pus pnqua$g '[gf] uralsdg pu€ rplrp^roS '[gt] €ql"uroC pue ollersqryse q?ns s.raded fueru ere areql 'eceds qnpotu eql Jo frlauoa3 aq? rod

'[ee-v]eare{ru"N pue '[ggl] uaprcN '[gg1] gqcnqey4l ees'a.rourJeqtrnJl '[gat] nrX ol raJarosp e \'lzgzJ'l'vzl1.rad1o,11pu€'[t?I] rnsew'[ot] '[te] '[eg] '[ze]sreg o] enp srluerul€erl rno 'hq] proJurn4 pue au3rleq fq pacnporlur lsrs seal xrpuadde stqlur uar.r3 saceJrns uue[uar]I pasolc go aceds rlnpou aqt Jo uotlecgtlceduoc eqtr,

seloN

saloN t9z

References

Books and Proceedings

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[A-2] Ahlfors, L. V. : Lectures on Quasiconformal Mappings, D. Van Nostrand,Princeton, New Jersey, 1966.

[A-3] Ahlfors, L. V. : Conformal Inaariants, McGraw-Hill, New York, 1973.

[A- ] Ahlfors, L. V. : Complet Analysis,3rd ed. McGraw-HilI, New York, 1979.

[A-5] Ahlfors, L. V. : Collected Papers, Vols. 1, 2, Birkhi.user, Boston, 1982.

[A-6] Ahlfors, L. V. and Sario, L. : Riemann Surfaces, Princeton University Press,Princeton, New Jersey, 1960.

[A-7] Ahlfors, L. V. et al. (eds.) : Aduances in the Theory of Riemann Surfaces, 1969Stong Brook Conference, Ann. Math. Studies, No. 66, Princeton UniversityPress, Princeton, New Jersey, 1971.

[A-8] Ahlfors, L. V. et al. (eds.) : Contributions to Analysis, A Collection of PapersDedicated to L. Bers, Academic Press, London, 1974.

[A-9 ] Arbare l lo , E . , Corna lba , M. , Gr i f i ths , P .A.and Har r is , J . : Geomet ry o lAlgebraic Curues, Vol. I , Springer-Verlag, Berl in and New York,1984.

[A-10] Baily, W. L., Jr. ; Introductorg Lectures on Automorphic Forrns, Iwanami-Shoten, Tokyo, and Princeton University Press, Princeton, New Jersey, 1973.

[A-11] Beardon, A. F. : The Geometry ol Discrete Groups, Springer-Verlag, Berlinand New York, 1983.

[A-12] Bers, L. : Topology, Courant Institute of Mathematical Science, New YorkUniversity Press, New York, 1956-1957.

[A-13] Bers, L. : Riemann Surfaces, Courant Institute of Mathematical Science, NewYork University Press, New York, 1957-1958.

[A-14] Bers, L. ; Introduction to Seaeral Compler Variables, Courant Institute ofMathematical Sciences, New York University Press, New York, 1964.

[A-15] Bers, L. et al. (eds.) : A Crash Course on l{Ieinian Groups, Lecture Notes inMath., Vol. 400, Springer-Verlag, Berlin and New York, 1974.

[A-16] Bers, L., John, F. and Schechter, M. : Partial Differential Equations, AmericanMathematical Society, Providence, Rhode Island, Providence, Rhode Island,Providence, Rhode Island, 1964.

[A-17] Besse, A. L. : Einstein Manitolds, Springer-Verlag, Berlin and New York, 1987.

[A-18] Birman, J. S. ; Braids, Linles, and Mapping Class Groups, Ann. Math. Studies,No. 82, Princeton University Press, Princeton, New Jersey, 1975.

'1161 'uopuoT 'ssard f,ruap"rv 'ecuatatuog lDuorlnlqruJ aflpuqutog

g26y tsuotycung cttldtoutolnv puD tdnotg ep$st1 , (.p") .f .L4 ,rtarrreg [If-vJ

'9961 'fasral a,ra11 'uolarurr4'ssar4 dlrsraaru1 uolaf,urrd 'ncoltng uuotxerA uo salnl?a1 : .C .U ,turuung

[g7-yJ.gl,6I ,{ro^

a,rag 'i(a116 'fitlatuoag cntqa1ly {o sa1dnut.t4 : .f (sur"H pu" .y .d ,sqfUIrC [6t-v]

'6961 'pue1s1

apoqg 'aouapr,ror4 ',(1aoog lpf,ll"rueql"I I uerrraruv ,gl.lo1 ,sqderEouo141

lecr-l"uraqlpl^l Jo suorl"Isu?rJ ,santng cntqa0yy ol uoq)n[nryul : .V

a ,sqfUIrD

[gt-y]'p161 'fasral alalq 'uolarurr4

(ssar4 r(lrsraarufl uolarurrd '62 'oN sarpnls .ql"W .uuv bcuataluog puol-frron EL6I

'eacoltng uuDuarA puo tdnotp snonurluocsr1 : .1 .traquaarg [lS-v] '886I 'pu"lsl apoqg 'acuapr,ror4 ',(laroog

Frrl"uaql"I4l uecrraury 'ruor1-oluatardag dnotg to frtgauoag : (.spa) .U 'V ,pp"tr[ pu€ .I^i .r14 ,ueurplog

[9t-vl'zg6I 'l"artuow 'l?arluow ap ?1lsra^run(T

ap sasserd sa1 'rlnrprronfi to taqndot4 ctlntalcDrDUC : .M .J ,turrqag

[9t-v]'lg6I '{ro

rrag'da11q't1otlua.laStq cqotponfi puo ffnaqa rellnuq)EJ: .g .g ,raurprng [le-Vl 'ZI6I 'I06I '168I 'lr"qllnts 'rauqnal 'D 'g '(Z 'I)

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ua{dtoutolny r?p uoeqtr eW r4!! ua0unnpoll : .g ,ura1y pu" .U ,arprrg [tt-vl

'I86I'{ro

rt eN puta uryrag '8epan-ra8urrdg 'saco{tng uuoraery uo sarnpel ; .g ,ralsrog [ZS-VJ '196I '{ro1 natri 'easlaqg ''pg po7 'tuotlcung ctydtouto1ny : 'U 'T 'prod ht-v]

tl6I '{ro ,rleN puu uqrag '8epan-raturrdg 'Zgt 'lo4

''qleIt ur saloN ernlf,aT 'tacottng uuDuatq uo suo.Icunl optlJ | .O .f ,r(eJ [gS-V]

'6tr61 'stre4 'aruetg ap anbr1eur9q1e141

?lg_r)os 'l,g-gg 'slo1 'anbsrrglsy 'arreururag fesr6 2261-9 261

'tacopng n7rns uo$rnqJ ap xnvaoil : .n ,nreuaod pue .g ,qcequapnsT ..V ,Hl"J

[62-V]'086I '{ro^ l|^aN

pue urlrag '3egan-raBurtd.g ,taco{tng uuDraatA : .1 ,rary pue .I I

.H ,szrlreJ [gz-v]

'(asaurdel) 9961'o{4o;'oqso;-of1o1

'tcttfit14 laal?ouaqpry lo quautdoleoa6, : (.spa) .1 ,uurrfoy pus .H ,pt l-zg ILZ-V]

'g961'atprrqure3'ssar4'lru11 atprrqureg'acodg cr1oqtadnn {o qcadty ".4auoeC puo 1oct1fi1ouy : (.pa) .g .q ,uralsdg

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'tdnotg uDrurelx puo fi6o7odoa louorcueurp-noq : (pa) .g .q ,u1e1sdS [SZ-V]

'886I '{IoA 1!leN Pu"urgag '3epan-raturrdg 'sarras

lggyq tepan-raBurrdg 'acuara;uoC IUSI I 9g6I

'II p,t" I 'slo1

llnpory puv cuoq)un.r utld.toutoyog : (.spa) .I" ?e .O ,urser( [tz-v]

'(lz6l'{ro^

n aN pu" uqrag 'tegan-ra8urrdg 'lurrdar) 096I

,{ro1,ta1q ,aruaosrc1u1,ncot

-rns IDurruW puo's0utddo141 Touttotuog ta1dnuu4 s.plqc.rr1 ; .g ,1uernog

[SZ-VJ 'Lg6I '{ro1 malq'ra,roq 'sacoltng uuvutetq uo iutddo1,y Totu.ro{uog :.11 ,uqog [ZZ-vJ

'086I 'uoPuorl Pu? {ro^

,rra11'sser4 urnuald'fitaaq1 aatng ca1dutog {o qooqdotcs y : .H .g ,suaura13 [IZ-VJ

'986I')IIo

i aN pue uqrag 'tepan-ra8urrdg '(taun7o11 lDtrouery q?noy 'g 'g) stofilouy

xaldutog puo fi4auoag ptyuetagtg : (.spa) .14i .11 ,seryeg pue .1 ,1a,rzqC [02-V]

'9961 'a8prrqureg (ssar6 ',rru1 atpuqureg'uo1ttnq7 puouesp,N nlto saco{tng {o tutsttldtouolnv : 'V .S 'rapalg pu? 'f .v ,uosseg

[6I-VJ

992sef,ueralau

256 References

[A-42] Hirsh, M. W. : Differential Topologg, Springer-Verlag, Berlin and New York,1976.

[A-43] Hitotsumatsu, S.: Theory of Analytic Functions of Seaeral Complex Variables,

Baihukan, Tokyo, 1960 (Japanese).

[A-aa] Hubbard, J. H. : ,9ur les Sections Analytiques de la Courbe Uniaerselle de

Teichmiller, Memoires of the American Mathematical Society Vol. 4, Number

166, American Mathematical Society, Providence, 1976.

[A-45] Iitaka, S., Ifeno, K. and Namikawa, Y.: Spirit of Descartes and Algebraic

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'g96l '{ro1 ,uag 'ra,roq 'rr1tor14 pepenoC : 'g 'uueurary [26-v)

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oqofiy 't11o1ty '1 'suorleunolsu"rl r"Inporu pue saceds raflSurqtrral uO : 'H 'earqg [1OZJ

'299-Ttq '(seor) tg'7

'tttoN

nqoqgl 'saceds ralllurqf,ratr pu" $lsrp-Issnb yo uorlezrral)"r"qC : 'H 'e8qg [967]

T,gr-t1'(lAOf ) V7 'atup olofiyllIDW 't'saceds ralllurq)ralJo sarlradord crrlauroat pue rrlflaue uO :'H'etqs [OOt]

'IIt-662 '(fSOf) I0g

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[86I].(asauedel) g?,2-g0z,(leot) se

nqoigg 'suorlrun3:urearls Iq saf,"Jrns uu"luary uado 1o uorl"zrleeg : 't{ teqgs [fOf]

'6II-86 '(egOf ) 8V'aPH

'Ulolr[ 'YeuuroC'sBurddrur ra[]urqf,reJ ulelraf,;o flradord lsuarlxa aqJ : 'C 'C 'sar"qlas

[g6t]'929-109 '(sgot) g'fii 'utoufiO puD 'ttJ, 'po6tg 'arnle,rrnc a,rrletau

lu"lsuof, Jo sa)"Jrns uo sersapoaB 1o turpoc ^o{r"I [ l€f,rrlauroeD : 'g 'sagag [96I]

'gg6t ''gZ-r9Z 'dd 'II 'IoA '[?Z-Vl ur 'suorlcun;: qfual

lsapoat i(q saceds rallBurqrlel Jo uotl"firlaru"r"d : 'I 'r1v,rrog pu? 'W 'epddag [rcIJ

'0lC-998 '(geOf) ge't 'tttory nqoqg;'pun1 puoces aq1;o sdnortu"rsqrnJJo sacuds ralnurqrral aqlJo np"rlno : '11 'oloureur" pup 'g'e,taet5r1ag

[C6t]

L9Zsaf,uaraJau

268 References

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[218] Sunada, T. : Riemannian coverings and isospectral manifolds, Ann. Math. L2L

(1985) , 169-186.

[219] Takhtadzhyan, L. A. : Uniformization,,local index theorem, and geometry ofthe moduli spaces of Riemann surfaces and vector bundles, in Theto Functions,Bowdoin 1987, Ehrenpreis, L. and Gunning, R. C. (eds.), Proceedings of Sym-posia in Pure Math. Vol. 49, Part I, pp. 581-596, American Mathematical Soci-

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Complet Variable, Komatu, Y., Niino, K., and Yang C. (eds.) Pitman Research

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[225] Taniguchi, M. : Abelian differentials with normal behavior and complex pinching

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[226] Taniguchi, M. : On the first variation of Green's functions under quasiconformal

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[227] Taniguchi, M. : A note on the second variational formulas of functionals on

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[230] Teichrniller, O. : Bestimmung der extremalen quasikonformen Abbildungen

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'ZLv-Ltv'(oOOr) rt,'uroaD 'filO 't'aarno

lesra,rrun aql Jo frlauroa8 aql pue rularu )rloqrad,{q aq; : 'y .g ,1radyo7y1 [OSZ]

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f ) Ztl'sfrUd 'qlory 'unutuop 'sare3:rns uueruerg 1o aeeds aqluo uorlf,unJ ulaz Sraqlag aql pu? runrlcads aql Jo sf,rlotdrudsy : .V .S ,fradloA\

[ZgZ]' 962-9 Lz'(leot) gz' arc?D

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'Igt-tzt '(oror) 60r'wvn 'uuv'saleyrns uu"uaru lczduroc roJ rlnpo.'' se erlrads {fua1 aq; : .y .S ,yaA1o6

[g?ZJ' LLg-tLg'(SfOt) Tg.rttory' 7 c$co7'aceds

rafl]urq)ral roJ rrrleru uossra]ad-lre6 aql Jo ssaualalduroc-uo1q : 'y 'g '1rad1o46 [976]

'6Lt-6r''(OgOt) 6Z'rucaC '#lO 't 'sdeur cruourreq;o ,(roaq1 ra[]ruqf,ral erII : 'W 'y1o16

[llZJ '(gS/lSOt) Ol.tpqrnog 'urag 'uueurarg ap sar"Jrns sap salnporu sal rns : 'V 'l-rad\ [gtz]

.Zt-IZ ,(086I) 711 .t6go1,,y .uuy'sanbrrlaurosr uou la salerlcadsosr sauueruu?ut[arr s?l?rre1 : 'J 'Il[ 'sergutrn

[6pg]'8nS

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[1pg]90I-t0I

'(ggot) vl 'S 'N 'V 'nng 'aceds ra11$ur{f,ral re^o scrureufq : 'y '1v\ 'qlea1 [otz] '0c9-Itt'(ggor) tzl'UIVW 'uuy 'noy rrsapoa8 rafllurqrral aqJ, : 'y'14 'qcaan [697]

't6I-t9I '(SeOf ) tgl 'qIDn opy 'dnor8 snlqgru " qt-r^r

alqrleduror s8urdderu crrlatuurrfsrsenb ;o uorsualxa l"urro]uof,rscn$ : '4 'eqn; [SSZJ 'i886I '{ro1 ,traN pu" uqrag '3epan-ra8urrdg '98I-99I 'dd '9991 '1on

'srrl"utrreqlpel I ur saloN ernlf,al '('p") 'n 'elurnberg 'uot1otto11

to rn1ncyog uttctdol ur tfroaql rafl]urqlral o1 qceordde ["uorl"rr"a pf,rss"If, v:'f

'v'equrorl [ggg]

' 092-6VZ' (tgO f ) 69' UlDut olducenuo yg'aceds rallnurqcra;uo f,rrlau uossraledlra1\ aql roJ uorlf,unJ ftraua u" uO : 'f 'v '"qurorJ

[gt3]

692sef,uaraJau

)70 References

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12651 Zograf., P. G. and Takhtadzhyan, L. A. : On uniformization of Riemann surfacesand the Weil-Petersson metric on Teichmiiller and Schottky spaces, Math. USSRSbornik 60 (1988), 297-313.

12661 Zograf., P. G. and Takhtadzhyan, L. A. : On the geometry of moduli spaces ofvector bundles over a Riemann surface, Math. USSR Izuestiga 35 (f 990), 83-100.

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List of Symbols

19319119920020120r727258

2021 8 , 7 8 , 1 2 5

184202184609392

18754

1931936872

22453

61, 68189

37, 141, 183183

o

24416

24616, 162

247t6284

189, 1917223

17, r241 7 , 1 8 , 8 8 , 1 2 5

153

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pp(a),t'(u),-rpiruJ

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[']ldl,t

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Index

A

absolutely continuous on lines, 77absolutely extremal selfmapping, 173ACL,77act properly discontinuously, 31admissible, 174Ahlfors'theorem, 202Ahlfors-Weill theorem, 153Ahlfors-Weill's section, 157almost complex structure, 202analytically f inite type (g,n),75attractive fixed point, 37Aut(X)-conjugate, 36axis, 38

Bbase point of Teichmiiller

space, 120Beltrami coefficient, 16, 17,92,

r24, r25Beltrami coefficient induced by

a Riemannian metric, 22Beltrami equation, 21Beltrami differential, 124Bergman projection, 188Bers cohomology class, 199Bers' Beltrami diferential. 153Bers 'embedding, 150Bers' extremal problem, 172Bers'f iber space, 180Bers 'pro ject ion, 150Bers' simultaneous uniformization, L47Bieberbach's area theorem, 152biholomorphic mapping, 2, I59,246biholomorphically equivalent, 2, 246Brower's theorem on invariance

of domains. 67

Index

cCalder6n-Zygmund's theorem, 96canonical base of the space of

holomorphic Abeliandifferentials, 236

canonical form, 36canonical homology base, 236canonical lift, 121canonical p-qc mapping of e , 102canonical ;r-qc mapping of H, 104canonical quasiconformal mapping

of C with complex dilatation pr,102canonical period matrix, 237canonical system of generators, 5, 47

Carath6odry distance, 180Cartan's theorem, 166(C, e)-small, 248closed geodesic corresponding

to C.r, 54closed geodesic corresponding

to 1, 54closed Riemann surface of genus g, 5closed Riemann surface of genus g

with nodes, 245coboundary, 197cocycle condition, 197colla.r lemma, 174,249complete, 168complex dilatation, 18, 88complex dynamics, 118complex structure

of a Riemann surface, 1o f C / f , 8o f D (R) ,249o f . R ,29ot fr,/r, szof Mo,166

16I'6,(q pacnpur

PrluereJrP rru"rllag ?ruol.ur€q

16I,d fq pacnpur

FlluaraJrp rurerllag sruoureq

I6I '89I

' prlua.ragrp rtu€rlleg cruorur"qgll'uorlrpuoc s(uo?lrueg

t8z'uorlerr€^ s(Pr?r.ueP"H

H

69 ,lualearnba-.7

gg1'uraloaql s.qrszlorC

tg7,'T,iT,'93'uorlcuny ueeJC

62'raqurnu uorlf,asrelur crJleruoaE

Ig (uorlcunJ

{l3ua1 orsapoa3pg 'rrsapoa3

Ug'arnlelJnc u"rssneC

c

I 0Z ( w toJ-Z l"luau€punJ

zt 'g 'uorleler

l"luauePunJ6p'ureruop lelueurepunJ

67 ,lapour uersqcnd

gp 'dnor3 u€rsqrnJg7 'a?"ds a{clrd

29'Surppaqura urely-e{rrrJ

97 ,saleurplooc e{?rrd

961 'Eurdderu rellnuqtrel purroJ

97,7,'PlpA rolte^ NJ616

,uorleuroJap Nd

911'arnlcnrls xalduroc leururu-;f961

'dnor3 ,(Solouoqoc relqtlg tsrs6lz' L9'uorl"turoJep uaslerNlarlcuad

gv7,'glz' gg '69 'saleurplooc uaslarNleq?ued

J

69 'adf1

leuorldacxe0g

(^roaql crpoS.ra

69 '.;r rapun lualearnba

'LVZ'7,91 'gtl 'gal '0zl 'Gt'62'tz'?I'gI'zr'

L'?,'lualearnbag'1el3e1ur cr1dr11e

t 'elrnc crldqla

ZLI 'Lg ,cr1dq1a

661 'sse1c fSolotuoqoc ralqcrg

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247,warcaq1 flqenp

96I (uraroeql s(llneaqloq

gg1'fl.radord Sursearcap ecuelsrp

tt '�elartsrP

'6I'seceJrnsuueruerg go flgreg elqerluareJrp

6ZI ,gg ,1sr,u,1 uqaq

1p6rsace;lns uueruarll 3o uorle.reua3ap

W7,' M,'6p6'uorleraua8app61 'arnlcnrls xaldtuoc Jo uorleuroJep

C

621 '1urod

leururur-X696'rosual alnl€Arnc

96'dnor3 uorleruJoJsuerl Eurraloe

96'uorleurrogsue.rl Surrarr,oc

lU (e?eJJns turrarroc

27'deru Surraaoc

/U'3uua^octg

'elrnc pasolJ € relo?

6 'd punore pooq.roqq3rau el€urproot

6'pooqloqqSrau aleurproocgt'(X)?nV ut ale3nfuoc

16' 6' lualuttrnba flpurro;uoc

IZ 'crrleul ueruu€ruerll e

dq pacnpur arn??nrls leruroJuocI6

'Z reJnlrnrls I€ruroJuoc

16 '6 'Surddeur

leuroJuof,gvz'6w Jowz'rw lo

uorlecgrlceduroc

69I 'I 'ploJluetu xalduroc

IgI .(J)J

JoI9I

'(u)J Jo

IgI ,6J

JOlgl

'lgr 'dJ Jo

T9I ,gJ

JO097,'6w Jo

xapul

. ! t o

harmonic map, 218Hartogs'theorem, 159Hermitian inner product

on ?|("(i-)), 200Ilermitian inner product

on {,("(r)),200holomorphic automorphic form

(of weight -:4), L28holomorphic family of Riemann

surfaces, 180, 194holomorphic function, 2, I59,249holomorphic mapping, 2, I59holomorphic quadratic differential,

73,128holomorphic sectional curvature, 210holomorphic tangent space

of ?( f ) , 189, 192horizontal trajectory, 142hyperbolic, 37, I72hyperbolic complex manifold, 168hyperbolic length, 53hyperbolic,L@-norm, 150hyperbolic metric, 54

I

improved ,\-lemma, 118infinitesimal deformation, 194initial differential, 140initial point, 28irreducible. 174isothermal coordinates. 20

J

Jacobi's problem, 253

KK2ihler metric, 202Klein-Maskit combination theorem,

250Klein's combination theorem, 65Kleinian group, 50, L79,217Kobayashi distance, 168Kobayashi pseudo-distance, 168

Index

Kodaira-Spencer deformationtheory, 181, 194

K-qc, 78, 120

L

lattice group, 7lie over a point p, 29lift of a mapping, 30lift of a path, 29, 30limit set, 44linear fractional transformation, 34local coordinate, 2local coordinate a.round p, 2local parameter, 2lbcal parameter around p, 2loxodromic, 37-LP-smoothing sequence, 84,\-lemma, 118

M

mapping class group of iR, 16, 162marked closed Riemann surface

of genus g, 14marked torus, 12marking, 12, 14Maskit coordinates, 179maximal.60maximal dilatation, 18, 78matrix representation, 35measurable automorphic form, 187measured foliation, 73Miibius transformation, 34modular group, 9moduli space of closed Riemann

surfaces ofgenus g, 16moduli space of tori, 9module, 84Mori's theorem, 92multiplier, 37Mumford's compactness theorem, 175

791,e1nu.ro; Surcnpolda.r

29 ,1urod paxg Burllada.r

gg'uo11caga.r

?lI 'elqrrnpar

771 'err-ds rellnuqcral pecnpar

tLl '{"C' ' ' ''rC} dq pacnpar

247, urc rcaq1 flrco.rdrcarI0Z'(J)Jyo aceds lua3uel pa.r

gg ,6 ae.t3ap;odnor3 reeurl lercads lear

?g (uotl€ruroJsusrl

snrqol{ Isarvt ' uorleur.rogsueJl leuorlc"{ r"eurl Iear

.296'elnurro; Ieuor?€rr"^ s(rlrneg

u

69 'g ,f3o1odo1

luarlonb69

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278

reproducing kernel, 184Ricci curvature, 209Ricci curvature tensor, 209Riemann sphere, 3Riemann surface, 1Riemann surface of type (g ,n), 75Riemann sur face of type (g,n,m),75

Riemann's mapping theorem, 25Riemannian curvature tensor, 209

Riemannian metric, 20Riemannian metric

correspondingto f ,23Riesz-Thorin's convex theorem, 116Royden's theorem, 168, 170

ssame complex structure, 2same conformal structure, 21scalar curvature, 210Schiffer's interior variation, 233Schiffer-Spencer's variat ion, 242Schiffer-Spencer's variation

by attaching a handle, 242Schifer-Spencer's variational

formula, 242Schottky group, 50Schottky space, 50Schwarzian derivative, 149Schwaz-Pick's lemma, 51Selberg zeta function, 253Serre's duality theorem, 196Shimizu's lemma, 45Siegel upper half-space, 237space of infinitesimal

deformations, 194special unitary group

o fs igna tu re (1 ,1 ) , 35straight line space, 176straight line, 176string theory,2I8,253strong deformation, 247strong deformation space, 247strongly equivalent, 23super Riemann surface, 253

Index

system of coordinate neighborhoods, 1system of decomposing curves, 60

T

Teichmiiller curve, 180Teichmiiller distance, I25, 162Teichmiiller mapping, 129Teichmiiller modular group, 16, 162Teichmiiller modular

transformation, 16, 162, L72Teichmiiller modular transformation

group, 247Teichmilller space

ofgenus 1, 12,2I4of genus g, 14, 127 ,of .r?, 13, 14,120, 2I5of a torus, 13of l , I22, 123, 148, 151

Teichmiiller's existence theorem, 134Teichmiiller's lemma, 138, 190Teichmiil ler's theorem. 59. 134Teichmiiller's uniqueness theorem, 132

terminal, 247terminal differential, 140terminal point, 283-manifold, 50Thurston's boundary, 75Thurston's compactifi cation, 75topology

o f C / f , 8o f D (R) ,248o f R , 2 9ot fr,/r, szo f Mo ,166

o f Mn ,246of Tn, 48of ?(.R), 48,125of T(f), 125

torus, 4trace, 37translation of the base point, 127, 159,

twisting parameter, 62d-horizontal line, 142

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