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MINIMAL SURFACES OF LEAST TOTAL CURVATUREAND MODULI SPACES OF PLANE POLYGONAL ARCS

Matthias WeberMathematisches Institut

Universitat BonnBeringstrae 6

Bonn, Germany

Michael Wolf*Dept. of Mathematics

Rice UniversityHouston, TX 77251 USA

Revised April 13, 1998

Introduction. The first major goal of this paper is to prove the existence of completeminimal surfaces of each genusp >1 which minimize the total curvature (equivalently, thedegree of the Gau map) for their genus. The genus zero version of these surfaces is knownas Ennepers surface (see [Oss2]) and the genus one version is due to Chen-Gackstatter([CG]). Recently, experimental evidence for the existence of these surfaces for genus p35was found by Thayer ([Tha]); his surfaces, like those in this paper, are hyperelliptic surfaceswith a single end, which is asymptotic to the end of Ennepers surface.

Our methods for constructing these surfaces are somewhat novel, and as their devel-opment is the second major goal of this paper, we sketch them quickly here. As in theconstruction of other recent examples of complete immersed (or even embedded) mini-mal surfaces inE3, our strategy centers around the Weierstra representation for minimalsurfaces in space, which gives a parametrization of the minimal surface in terms of mero-morphic data on the Riemann surface which determine three meromorphic one-forms onthe underlying Riemann surface.

The art in finding a minimal surface via this representation lies in finding a Riemannsurface and meromorphic data on that surface so that the representation is well-defined,

i.e., the local Weierstra representation can be continued around closed curves withoutchanging its definition. This latter condition amounts to a condition on the imaginaryparts of some periods of forms associated to the original Weierstra data.

In many of the recent constructions of complete minimal surfaces, the geometry of thedesired surface is used to set up a space of possible Weierstra data and Riemann surfaces,

*Partially supported by NSF grant number DMS 9300001 and the Max Planck Institut; Alfred P. Sloan

Research Fellow

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and then to consider the period problem as a purely analytical one. This approach is veryeffective as long as the dimension of the space of candidates remains small. This happensfor instance if enough symmetry of the resulting surface is assumed so that the modulispace of candidate (possibly singular) surfaces has very small dimension in fact, thereis sometimes only a single surface to consider. Moreover, the candidate quotient surfaces

(for instance, a thrice punctured sphere) often have a relatively well-understood functiontheory which serves to simplify the space of possibilities, even if the (quite difficult) periodproblem for the Weierstra data still remains.

In our situation however, the dimension of the space of candidates grows with thegenus. Our approach is to first view the periods and the conditions on them as defininga geometric object (and inducing a construction of a pair of Riemann surfaces), and tothen prove analytically that the Riemann surfaces are identical, employing methods fromTeichmuller theory.

Generally speaking, our approach is to construct two different Riemann surfaces, eachwith a meromorphic one-form, so that the period problem would be solved if only thesurfaces would coincide. To arrange for a situation where we can simultaneously define aRiemann surface, and a meromorphic one-form on that surface with prescribed periods,we exploit the perspective of a meromorphic one-form as defining a singular flat structureon the Riemann surface, which we can develop onto E2.

In particular, we first assume sufficient symmetry of the Riemann surface so that thequotient orbifold flat structure has a fundamental domain in E2 which is bounded by aproperly embedded arc composed of 2p+ 2 horizontal and vertical line segments withthe additional properties that the segments alternate from horizontal segments to verticalsegments, with the direction of travel also alternating between left and right turns. Wecall such an arc a zigzag; further we restrict our attention to symmetric zigzags, thosezigzags which are symmetric about the line{y= x} (see Figure 1).

A crucial observation is that we can turn this construction around. Observe that azigzag Zbounds two domains, one, NE(Z), on the northeast side, and one, SW(Z), onthe southwest side. When we double each of these domains and then take a double coverof the resulting surface, branched over each of the images of the vertices ofZ, we havetwo hyperelliptic Riemann surfaces,RNE(Z) andRSW(Z), respectively. Moreover, theformdz when restricted to NE(Z) and SW(Z), lifts to meromorphic one-formsNE(Z)on NE(Z) and SW(Z) on SW(Z) both of whose sets of periods are integral linearcombinations of the periods ofdz along the horizontal and vertical arcs ofZ.

Then, suppose for a moment that we can find a zigzag Zso that NE(Z) is conformallyequivalent to SW(Z) with the conformal equivalence taking vertices to vertices (where

is considered a vertex). (We call such a zigzag reflexive.) Then

RNE(Z) would be

conformally equivalent toRSW(Z) in a way that ei/4NE(Z) and ei/4SW(Z) haveconjugate periods. As these forms will represent gdh and g1dh in the classical Weier-

stra representation X = Re

( 12

g 1g

, i2

g+ 1g

, 1) dh, it will turn out that this

conformal equivalence is just what we need for the Weierstra representation based onNE(Z) and SW(Z) to be well-defined.

This construction is described precisely in3.2


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We are left to find such a zigzag. Our approach is non-constructive in that we considerthe spaceZp of all possible symmetric zigzags with 2p+ 2 vertices and then seek, withinthat spaceZ, a symmetric zigzag Z0 for which there is a conformal equivalence betweenNE(Z0) and SW(Z0) which preserves vertices. The bulk of the paper, then, is an analysisof this moduli space

Zp and some functions on it, with the goal of finding a certain fixed

point within it.Our methods, at least in outline, for finding such a symmetric zigzag are quite standard

in contemporary Teichmuller theory. We first find that the spaceZp is topologically acell, and then we seek an appropriate height function on it. This appropriate heightfunction should be proper, so that it has an interior critical point, and it should have thefeature that at its critical point Z0 Zp, we have the desired vertex-preserving conformalequivalence between NE(Z0) and SW(Z0).

One could imagine that a natural height function might be the Teichmuller distancebetweenRNE(Z) andRSW(Z), but it is easy to see that there is a family Zt of zigzags,some of whose vertices are coalescing, so that the Teichmuller distance betweenRNE(Zt)andRSW(Zt) tends to a finite number. We thus employ a different height functionD()that, in effect, blows up small scale differences betweenRNE(Zt) and RSW(Zt) for a familyof zigzags{Zt} that leave all compacta ofZp.

We discuss the spaceZp of symmetric zigzags, and study degeneration in that space in4. We show that the map between the marked extremal length spectra forRNE(Z) andRSW(Z) is not real analytic at infinity in Zp, and thus there must be small scale differencesbetween those extremal length spectra. We do this by first observing that both extremallengths and Schwarz-Christoffel integrals can be computed using generalized hypergeomet-ric functions; we then show that the well-known monodromy properties of these functionslead to a crucial sign difference in the asymptotic expansions of the Schwarz-Christoffelmaps at regular singular points. Finally, these sign differences are exploited to yield the

desired non-analyticity.Our height function D(Z) while not the Teichmuller distance betweenRNE(Z) and

RSW(Z), is still based on differences between extremal lengths on those surfaces, and in ef-fect, we follow the gradient flowdDon Zpfrom a convenient initial point in Zpto a solutionof our problem. There are two aspects to this approach. First, it is especially convenientthat we know a formula ([Gar]) for d[Ext[](R)] where Ext[](R) denotes the extremallength of the curve family [] on a given Riemann surfaceR. This gradient of extremallength is given in terms of a holomorphic quadratic differential 2[](R) = d Ext[](R)and can be understood in terms of the horizontal measured foliation of that differentialproviding a direction field onRalong which to infinitesimally deformRas to infinitesi-mally increase Ext[](R). We then show that grad D() Zcan be understood in terms ofa pair of holomorphic quadratic differentials onRNE(Z) andRSW(Z), respectively, whose(projective) measured foliations descend to a well-defined projective class of measured fo-liations on C= NE(Z) Z SW(Z). This foliation class then indicates a direction inwhich to infinitesimally deform Zso as to infinitesimally decrease D(), as long asZis notcritical for D(). Thus, a minimum for D() is a symmetric zigzag Z0 for which NE(Z0)is conformally equivalent to SW(Z) in a vertex preserving way. Second, it is technically

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convenient to flow along a path inZp in which the form of the height function simplifies.In fact, we will flow from a genus p 1 solution inZp1 Z along a pathY inZp toa genus p solution inZp. Here the technicalities are that some Teichmuller theory andthe symmetry we have imposed on the zigzags allow us to invoke the implicit functiontheorem at a genus p

1 solution in

Zto find such a good path

Y Zp. Thus, formally,

we find a solution in each genus inductively, by showing that given a reflexive zigzag ofgenus p 1, we can add a handle to obtain a solution of genus p.

We study the gradient flow of the height function D() in5.Combining the results in sections 4 and 5, we conclude

Main Theorem B. There exists a reflexive symmetric zigzag of genusp forp0 whichis isolated inZp.

When we interpret this result about zigzags as a result on Weierstra data for minimallyimmersed Riemann surfaces in E3, we derive as a corollary

Main Theorem A. For eachp0, there exists a minimally immersed Riemann surfacesinE

3

with one Enneper-type end and total curvature4(p +1). This surface has at mosteight self-isometries.

In6, we adapt our methods slightly to prove the existence of minimally immersedsurfaces of genus p(k 1) with one Enneper-type end of winding order 2k1: thesesurfaces extend and generalize examples of Karcher ([Kar]) and Thayer ([Tha]), as well asthose constructed in Theorem A.

While we were preparing this manuscript several years ago we received a copy of apreprint by K. Sato [S] which also asserts Theorem A. Our approach is different than thatof Sato, and possibly more general, as it is possible to assign zigzag configurations to anumber of families of putative minimal surfaces. We discuss further applications of this

technique in a forthcoming paper [WW].The authors wish to thank Hermann Karcher for many hours of pleasant advice.

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2. Background and Notation.2.1 Minimal Surfaces and the Weierstra Representation. Here we recall somewell-known facts from the theory of minimal surfaces and put our result into context.

Locally, a minimal surface can always be described by Weierstra data, i.e. there arealways a simply connected domainU, a holomorphic functiong and a holomorphic 1-formdhinUsuch that the minimal surface is locally given by

zRe z

12 (g 1g )dhi2 (g+

1g )dh

dh

For instance,g(z) = z anddh = dzz will lead to the catenoid, whileg(z) =z anddh = zdzyields the Enneper surface.

It is by no means clear how global properties of a minimal surface are related to thislocal representation.

However, two global properties together have very strong consequences on the Weier-stra data. One is the metrical completeness, and the other the total (absolute) Gauiancurvature of the surface R, defined by

K:=R

|K|dA= 4 degree of the Gau map

We will call a complete minimal surface of finite absolute Gauian curvature a finiteminimal surface.

Then by a famous theorem of R. Osserman, every finite minimal surface (see [Oss1,Oss2, Laws]) can be represented by Weierstra data which are defined on a compactRiemann surface R, punctured at a finite number of points. Furthermore, the Weierstradata extend to meromorphic data on the compact surface. Thus, the construction of suchsurfaces is reduced to finding meromorphic Weierstra data on a compact Riemann surfacesuch that the above representation is well defined, i.e. such that all three 1-forms showingup there have purely imaginary periods. This is still not a simple problem.

From now on, we will restrict our attention to finite minimal surfaces.Looked at from far away, the most visible parts of a finite minimal surface will be the

ends. These can be seen from the Weierstra data by looking at the singularitiesPj of theRiemannian metric which is given by the formula

(2.0) ds= |g| + 1

|g| |dh|An end occurs at a puncture Pj if and only if ds becomes infinite in the compactifiedsurface at Pj .

To each end is associated its winding or spinning number dj , which can be definedgeometrically by looking at the intersection curve of the end with a very large spherewhich will be close to a great circle and by taking its winding number, see [Gack, J-M].

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This winding number is always odd: it is 1 for the catenoid end 3 for the Enneper end.There is one other end of winding number 1, namely the planar end which of course occursas the end of the plane, but also as one end of the Costa surface, see [Cos1]. For finiteminimal surfaces, there is a Gau-Bonnet formula relating the total curvature to genusand winding numbers:

R

KdA = 2

2(1 p) r rj=1

dj

where p is the genus of the surface, r the number of ends and dj the winding number ofan end. For a proof, see again [Gack, J-M].

From this formula one can conclude that for a non planar surfaceR |K|dA 4(p + 1)

which raises the question of finding for each genus a (non-planar) minimal surface forwhich equality holds. This is the main goal of the paper.

The following is known:

p= 0: The Enneper surface and the catenoid are the only non-planar minimal surfaceswithK= 4, see e.g. [Oss2].

p= 1: The only surface withK = 8 is the Chen-Gackstatter surface (which is definedon the square torus), see [CG, Lop, Blo].

p= 2: An example withK= 12 was also constructed in [CG]. Uniqueness is not knownhere.

p= 3: An example withK= 16 was constructed by do Esprito-Santo ([Esp]).p35: E. Thayer has solved the period problem numerically and produced pictures of

surfaces with minimalK.Note that all these surfaces are necessarily not embedded: For given genus, a finite

minimal surface of minimal

Kcould, by the winding number formula, have only either one

end of Enneper type (winding number 3) which is not embedded or two ends of windingnumber 1. But by a theorem of R. Schoen ([Sch]), an embedded finite minimal surfacewith only two ends has to be the catenoid. Hence if one looks for embedded minimalsurfaces, one has to allow moreK. For the state of the art here, see [Ho-Ka].

If one allows even more total curvature and permits non-embeddedness, some generalmethods are available, as explained in [Kar].

2.2. Zigzags. AzigzagZof genuspis an open and properly embedded arc in Ccomposedof alternating horizontal and vertical subarcs with angles of/2, 3/2,/2, 3/2, . . . , /2between consecutive sides, and having 2p+ 1 vertices (2p+ 2 sides, including an initialinfinite vertical side and a terminal infinite horizontal side.) A symmetriczigzag of genus

pis a zigzag of genus p which is symmetric about the line{y= x}. The spaceZp of genuspzigzags consists of all symmetric zigzags of genus pup to similarity; it is equipped withthe topology induced by the embedding ofZpR2p which associates to a zigzag Z the2p-tuple of its lengths of sides, in the natural order.

A symmetric zigzag Zdivides the plane C into two regions, one which we will denoteby NE(Z) which contains large positive values of{y = x}, and the other which we willdenote by SW(Z). (See Figure 1.)

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P0

P1

P-1

Pp

P-p

NE

SW

Figure 1

Definition 2.2.1. A symmetric zigzag Z is called reflexive if there is a conformal map: NE(Z)SW(Z) which takes vertices to vertices.Examples 2.2.2. There is only one zigzag of genus 0, consisting of the positive imaginaryand positive real half-axes. It is automatically symmetric and reflexive.

Every symmetric zigzag of genus 1 is also automatically reflexive.

2.3. Teichmuller Theory. For M a smooth surface, let Teich (M) denote the Teich-muller space of all conformal structures on M under the equivalence relation given bypullback by diffeomorphisms isotopic to the identity map id: M M. Then it is well-known that Teich (M) is a smooth finite dimensional manifold ifMis a closed surface.

There are two spaces of tensors on a Riemann surfaceR that are important for theTeichmuller theory. The first is the space QD(R) of holomorphic quadratic differentials,i.e., tensors which have the local form = (z)dz2 where (z) is holomorphic. Thesecond is the space of Beltrami differentials Belt(R), i.e., tensors which have the localform= (z)dz/dz.

The cotangent space T[R](Teich (M)) is canonically isomorphic to QD(R), and thetangent space is given by equivalence classes of (infinitesimal) Beltrami differentials, where

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1 is equivalent to 2 ifR

(1 2) = 0 for every QD(R).

If f : C C is a diffeomorphism, then the Beltrami differential associated to thepullback conformal structure is = fzfz

dzdz . If f is a family of such diffeomorphisms

with f0 = identity, then the infinitesimal Beltrami differential is given by dd

=0

f =dd

=0

fz

. We will carry out an example of this computation in5.2.A holomorphic quadratic differential comes with a picture that is a useful aid to ones

intuition about them. The picture is that of a pair of transverse measured foliations,whose properties we sketch briefly (see [FLP], [Ke], and [Gar] for more details). We nextdefine a measured foliation on a (possibly) punctured Riemann surface; to set notation, inwhat follows, the Riemann surfaceRis possibly punctured, i.e. there is a closed RiemannsurfaceR, and a set of points{q1, . . . , q m}, so thatR=R {q1, . . . , q m}.

A measured foliation (F, ) on a Riemann surfaceR with singularities{p1, . . . , pl}(where some of the singularities might also be elements of the puncture set {q1, . . . , q m})consists of a foliation FofR{p1, . . . , pl}and a measure as follows. If the foliation F isdefined via local charts i : Ui R2 (where{Ui}is a covering ofR{p1, . . . , pl}) whichsend the leaves ofFto horizontal arcs in R2, then the transition functions ij :i(Ui)j(Uj) on i(Ui) R2 are of the form ij(x, y) = (h(x, y), c y); here the function his an arbitrary continuous map, but c is a constant. We require that the foliation in aneighborhood (inR) of the singularities be topologically equivalent to those that occur atthe origin in Cof the integral curves of the line fieldzkdz2 >0 wherek 1. (There areeasy extensions to arbitrary integral k , but we will not need those here.)

We define the measure on arcs A

Ras follows: the measure (A) is given by

(A) =

A

|dY|

where|dY| is defined, locally, to be the pullback|dY|Ui = i (|dy|) of the horizontaltransverse measure|dy| on R2. Because of the form of the transition functions ij above,this measure is then well-defined on arcs inR.

An important feature of this measure (that follows from its definition above) is itstranslation invariance. That is, suppose A0 R is an arc transverse to the foliationF, with A0 a pair of points, one on the leafl and one on the leafl; then, if we deformA0 toA1 via an isotopy through arcs At that maintains the transversality of the image ofA0 at every time, and also keeps the endpoints of the arcs At fixed on the leaves l and l

,respectively, then we observe that (A0) =(A1).

Now a holomorphic quadratic differential defines a measured foliation in the followingway. The zeros 1(0) of are well-defined; away from these zeros, we can choose a

canonical conformal coordinate (z) =z

so that = d2. The local measuredfoliations ({Re = const}, |d Re |) then piece together to form a measured foliation known

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as the vertical measured foliation of , with the translation invariance of this measuredfoliation of following from Cauchys theorem.

Work of Hubbard and Masur ([HM]) (see also alternate proofs in [Ke], [Gar] and [Wo]),following Jenkins ([J]) and Strebel ([Str]), showed that given a measured foliation (F, )and a Riemann surface

R, there is a unique holomorphic quadratic differential on

Rso that the horizontal measured foliation of is equivalent to (F, ).Extremal length. The extremal length ExtR([]) of a class of arcs on a Riemann

surfaceRis defined to be the conformal invariant

sup

2()

Area()

where ranges over all conformal metrics onR with areas 0 < Area()


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An extraordinary fact about this metric is that the extremal maps, known as Teich-muller maps, admit an explicit description, as does the family of maps which describe ageodesic.

Teichmullers theorem asserts that ifR1 andR2 are distinct points in Tg, then thereis a unique quasiconformal h :

R1

R2 with h homotopic to the identity on M which

minimizes the maximal dilatation of all such h. The complex dilatation of h may bewritten(h) =k q|q| for some non-trivial qQD(R1) and some k, 0< k


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3. From Zigzags to minimal surfaces.Let Z be a zigzag of genus p dividing the plane into two regions NE and SW. We

denote the vertices of NEconsecutively byPp, . . . , P p and setP=. The vertices ofSWhowever are labeled in the opposite order Qj :=Pj andQ=. We double bothregions to obtain punctured spheres SNEandSSWwhose punctures are also called Pj and

Qj . Finally we take hyperelliptic coversRNE over SNE, branched over the Pj , andRSWover SSW, branched over the Qj , to obtain two hyperelliptic Riemann surfaces of genus

p, punctured at the Weierstra points which will still be called Pj and Qj . The degree 2maps to the sphere are called NE:RNESNE and SW:RSWSSW.Example 3.1. For a genus 1 zigzag, the Riemann surfacesRNE andRSWwill be squaretori punctured at the three half-period points and the one full-period point.

Now suppose that the zigzag Z is reflexive. Then there is a conformal map : NESW such that (Pj) = Qj . Clearly lifts to conformal maps : SNE SSW and:RNE RSW which again take punctures to punctures.

The surfaceR

NE will be the Riemann surface on which we are going to define theWeierstra data. The idea is roughly as follows: If we look at the Weierstra data for theEnneper surface, it is evident that the 1-forms gdh and 1gdh have simpler divisors than

their linear combinations which actually appear in the Weierstra representation as thefirst two coordinate differentials. Thus we are hunting for these two 1-forms, and we wantto define them by the geometric properties of the (singular) flat metrics on the surface forwhich they specify the line elements, because this will encode the information we need tosolve the period problem.

To do this, we look at the flat metrics onRNEandRSWwhich come from the followingconstruction. First, the domains NE and SW obviously carry the flat euclidean metric(ds=|dz|) metrics. Doubling these regions defines flat (singular) metrics on the spheresSNEand SSW(with cone points at the lifts of the vertices of the zigzags and cone angles ofalternately and 3/2). These metrics are then lifted toRNE andRSWby the respectivecovering projections.

The exterior derivatives of the multivalued developing maps define single valued holo-morphic nonvanishing 1-formsNEonRNEandSWonRSW, because the flat metrics onthe punctured surfaces have trivial linear holonomy. Furthermore, the behavior of these1-forms at a puncture is completely determined by the cone angle of the flat metric at thepuncture. Indeed, in a suitable local coordinate, the developing map of the flat metricnear a puncture with cone angle 2k is given by zk. Hence the exterior derivative of thedeveloping map will have a zero (or pole) of order k 1 there. Note that these consider-ations are valid for the point P as well if we allow negative cone angles. All this is well

known in the context of meromorphic quadratic differentials, see [Str].

Examples 3.2. For the genus 0 zigzag, we obtain a 1-form NE on the sphereRNE witha pole of order 2 at P which we can call dz and a 1-form SW on the sphereRSW witha zero of order 2 at P0 and a pole of order 4 at P which then is z

2dz.For a symmetric genus 1 zigzag,NEwill be closely related to the Weierstra-function

on the square torus, as it is a 1-form with double zero in P0 and double pole at P.11


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Furthermore, SW is a meromorphic 1-form with double order zeroes in P1 and fourthorder pole at P which also can be written down in terms of classical elliptic functions.

In general, we can write down the divisors of our meromorphic 1-forms as:

(NE) =P20 P22 P24 . . . P2(p1) P2 p odd

P21 P23 . . . P2(p1) P2 p even

(SW) =

Q21 Q23 . . . Q2p Q4 p oddQ20 Q22 Q24 . . . Q2p Q4 p even

(dNE) =P0 P1 P2 . . . Pp P3

Now denote by := ei/4

NE, :=e

i/4

SWanddh := const

dNE, where we

choose the constant such that = dh2 which is possible because the divisors coincide.Now we can write down the Gau map of the Weierstra data onRNE as g = dh , and wecheck easily that the line element (2.0) is regular everywhere onRNE except at the lift ofP.

One can check that the thus defined Weierstra data coincide for the reflexive genus0 and genus 1 zigzags with the data for the Enneper surface and the Chen-Gackstattersurface.

We can now claim

Theorem 3.3. IfZis a symmetric reflexive zigzag of genus p, then (

RNE, g, dh) as above

define a Weierstra representation of a minimal surface of genus p with one Enneper-typeend and total curvature4(p + 1).

Proof. The claim now is that the 1-forms in the Weierstra representation all have purelyimaginary periods. For dh this is obvious, because the form dh is even exact and so allperiods even vanish. Because of (g 1g )dh = and i(g+ 1g )dh = i(+ ) this isequivalent to the claim that and have complex conjugate periods. To see this, we firstconstruct a basis for the homology onRNEand then compute the periods of and usingtheir geometric definitions.

To define 2p cycles Bj onRNE, we take curves bj in NE connecting a boundarypoint slightly to the right of Pj+1 with a boundary point slightly to the left of Pj forj =p, . . . , p1. We double this curve to obtain a closed curve Bj onSNE whichencircles exactly Pj and Pj+1. These curves have closed lifts Bj toRNE and form ahomology basis. Now to compute a period of our 1-forms, observe that a period is nothingother than the image of the closed curve under the developing map of the flat metric whichdefines the 1-form. But this developing map can be read off from the zigzag one onlyhas to observe that developing a curve around a vertex (regardless whether the angle there

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is 2 or 32 ) will change the direction of the curve there by 180

. Doing this yieldsBj

=

Bj

ei/4 NE= 2ei/4 (Pj Pj+1)

Bj = Bj ei/4 SW= (Bj) ei/4 SW= 2ei/4 (Qj Qj+1)= 2ei/4 (Pj Pj1)

which yields the claim by the symmetry of the zigzag.Finally, we have to compute the total absolute curvature of the minimal surface. By

the definition of the Gau map we have

(g) =

P+10 P11 P+12 . . . P1p P p oddP10 P+11 P12 . . . P1p P p even

and thus deg g= p+ 1 which implies R KdA=4(p + 1) as claimed. Remark 3.4. We close by making some comments on the amount of symmetry involvedin this approach. Usually in the construction of minimal surfaces the underlying Riemannsurface is assumed to have so many automorphisms that the moduli space of possibleconformal structures is very low dimensional (in fact, it consists only of one point in manyexamples). This helps solving the period problem (if it is solvable) because this will thenbe a problem on a low dimensional space. In our approach, the dimension of the modulispace grows with the genus, and the use of symmetries has other purposes: It allows us toconstruct for given periods a pair of surfaces with one 1-form on each which would solvethe period problem if only the surfaces would coincide. Indeed we observe

Lemma 3.5. The minimal surface of genusp constructed below has only eight isometries,and at most eight conformal or anticonformal automorphisms that fix the end, indepen-dently of genus.

Proof. Observe that as the end is unique, any isometry of the minimal surface fixes theend. As the isometry necessarily induces a conformal or anti-conformal automorphism, it issufficient to prove only the latter statement of the lemma. Because of the uniqueness of thehyperelliptic involution, this automorphism descends to an automorphism of the puncturedsphere which fixes the image of infinity and permutes the punctures (the images of theWeierstra points). As there are at least three punctures, all lying on the real line, wesee that the real line is also fixed (setwise). After taking the reflection (an anti-conformalautomorphism) of the sphere across the real line, which fixes all the punctures, we are leftto consider the conformal automorphisms of the domain NE. Finally, this domain hasonly two conformal symmetries, the identity and the reflection about the diagonal. Thelemma follows by counting the automorphisms we have identified in the discussion.

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4. The Height Function on Moduli Space.4.1 The space of zigzagsZp and a natural compactification Z. We recall thespaceZp of equivalence classes of symmetric genus p zigzags constructed in Section 2.2;here the equivalence by similarity was defined so that two zigzags Z and Z would be

equivalent if and only if both of the pairs of complementary domains (NE(Z), NE(Z

))and (SW(Z), SW(Z)) were conformally equivalent. Label the finite vertices of the zigzag

byPp, . . . , P 0, . . . , P p. Thus, we may choose a unique representative for each class inZpby setting the vertices P0 {y = x}, Pp = 1, Pp = i and Pk = iPk for k = 0, . . . , p;here all the verticesPp, . . . , P p are required to be distinct. The topology ofZp defined inSection 2.2 then agrees with the topology of the space of canonical representatives inducedby the embedding ofZpCp1 byZ(P1, . . . , P p1). With these normalizations andthis last remark on topology, it is evident thatZp is a cell of dimension p 1.

We have interest in the natural compactification of this cell, obtained by attaching aboundaryZpto Z. This boundary will be composed of zigzags where some proper consec-utive subsets of

{P0, . . . , P p

}(and of course the reflections of these subsets across

{y= x

})

are allowed to coincide; the topology onZp =Zp Zp is again given by the topology onthe map of coordinates of normalized representatives Z Zp(P1, . . . , P p1)Cp1.

Evidently, Zp is stratified by unions of zigzag spacesZkp of real dimension k, witheach component ofZkp representing the (degenerate) zigzags that result from allowing kdistinct vertices to remain in the (degenerate) zigzag after some points P0, . . . , P p havecoalesced. For instanceZ0p consists of the zigzags where all of the points P0, . . . , P p havecoalesced to either P0 {y =x} or P1 = 1, and the facesZp2p are the loci inZp whereonly two consecutive vertices have coalesced.

Observe that each of these strata is naturally a zigzag space in its own right, and onecan look for a reflexive symmetric zigzag of genus k+ 1 withinZkp .

4.2 Extremal length functions onZ. Consider the punctured sphereSNEin 3, wherewe labelled the punctures Pp, . . . , P 0, . . . , P p, and P and observed that SNE had tworeflective symmetries: one about the image of Z and one about the image of the curve{y = x} on NE. Let [Bk] denote the homotopy class of simple curves which enclosesthe punctures Pk and Pk+1 for k = 1, . . . , p 1 and [Bk] the homotopy class of simplecurves which encloses the punctures Pk and Pk1 fork= 1, . . . , p 1. Let [k] denotethe pair of classes [Bk] [Bk]. Under the homotopy class of maps which connectsSNEtoSSW (lifted from : NESW, the vertex preserving map), there are correspondinghomotopy classes of curves on SSW, which we will also label [k].

SetENE

(k) = ExtSNE

([k

]) andESW

(k) = ExtSSW

([k

]) denote the extremal lengths of[k] in SNE and SSW, respectively.

Let Tsymm0,2p+2 denote a subspace of the Teichmuller space of 2p+ 2 punctured sphereswhose points are equivalence classes of 2p+2 punctured spheres (with a pair of involutions)coming from one complementary domain NE(Z) of a symmetric zigzagZ. ThisT

symm0,2p+2is

ap 1 dimensional subspace of the Teichmuller spaceT0,2p+2of 2p +2 punctured spheres.Consider the map ENE: T

symm0,2p+2Rp1+ given bySNE(ENE(1), . . . , E NE(p 1)).

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Proposition 4.2.1. The map ENE: Tsymm0,2p+2Rp1+ is a homeomorphism ontoRp1+ .

Proof. It is clear that ENE is continuous. To see injectivity and surjectivity, apply aSchwarz-Christoffel map SC : NE {Im z > 0} to NE; this map sends NE to theupper half-plane, taking Z to R so that SC(P) =, SC(P0) = 0, SC(P1) = 1 andSC(P

k) =

SC(Pk). These conditions uniquely determine SC; moreover E

NE(k) =

2ExtH2(k) where k is the class of pairs of curves in H2 that connect the real arc

between SC(Pk2) and SC(Pk1) to the real arc between SC(Pk) and SC(Pk+1),and the arc between S C(Pk1) andS C(Pk) to the arc between S C(Pk+1) andS C(Pk+2).Now, any choice ofp1 numbersxi= S C(Pi) for 21puniquely determines a point inTsymm0,2p+2, and these choices are parametrized by the extremal lengths ExtH2(k)(0, ).This proves the result.

Let QDsymm(SNE) denote the vector space of holomorphic quadratic differentials onSNE which have at worst simply poles at the punctures and are real along the image ofZand the line{y= x}.

Our principal application of Proposition 4.2.1 is the following

Corollary 4.2.2. The cotangent vectors{dENE(k)| k = 1, . . . , p 1} (and{dESW(k)|k= 1, . . . , p 1}) are a basis forTSNETsymm0,2p+2, hence forQDsymm(SNE).Proof. The cotangent space TSNET

symm0,2p+2 to the Teichmuller space T

symm0,2p+2 is the space

QD(SNE) of holomorphic quadratic differentials on SNE with at most simple poles at thepunctures. A covector cotangent toTsymm0,2p+2 must respect the reflective symmetries of the

elements ofTsymm0,2p+2, hence its horizontal and vertical foliations must be either parallel orperpendicular to the fixed sets of the reflections. Thus such a covector, as a holomorphicquadratic differential, must be real on those fixed sets, and hence must lie in QDsymm(SNE).The result follows from the functions

{ENE(k)

|k = 1, . . . , p

1

}being coordinates for

Tsymm

0,2p+2.

4.3 The height function D(Z) :Z R. Let the height function D(Z) be

(4.1) D(Z) =

p1j=1

exp

1

ENE(j)

exp

1

ESW(j)

2+ [ENE(j) ESW(j)]2 .

We observe that D(Z) = 0 if and only ifENE(j) = ESW(j), which holds if and only ifSNEisconformally equivalent toSSW. We also observe that, for instance, ifENE(j)/ESW(j)C0but both ENE(j) and ESW(j) are quite small, then D(Z) is quite large. It is this latter

fact which we will exploit in this section.

4.4 Monodromy Properties of the Schwarz-Christoffel Maps. Here we derive thefacts about the Schwarz-Christoffel maps we need to prove properness of the height func-tion.

Let t= (0< t1 < t2 < . . . < tp) be p points on the real line. We put t0 := 0, t:=andtk =tk. Then the Schwarz-Christoffel formula tells us that we can map the upper

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half plane conformally to a NE-domain such that the{ti} are mapped to vertices by thefunction

f(z) =

z0

(t tp)1/2(t tp+1)1/2 (t tp)1/2dt

and to a SW-domain by

g(z) =

z0

(t tp)1/2(t tp+1)+1/2 (t tp)1/2dt

Note that the exponents alternate sign. We are not interested in normalizing thesemaps at the moment by introducing some factor, but we have to be aware of the fact thatscaling the tk will scale f and g .

Now introduce the periods ak = f(tk+1) f(tk) and bk = g(tk+1) g(tk) which arecomplex numbers, either real or purely imaginary. Denote by

T :={

t: ti

C, tj

=tk

j, k}

,

the complex-valued configuration space for the 2p+ 1-tuples{t}. It is clear that we cananalytically continue theakand bkalong any path inTto obtain holomorphic multi-valuedfunctions.

Lemma 4.4.1. Continue ak analytically along a path in T defined by moving tj anti-clockwise aroundtj+1 and denote the continued function byak, similarly forbk. Then wehave

ak =

ak ifk=j 1, j+ 1ak+ 2ak+1 ifj =k+ 1

ak

2ak1 ifj =k

1

with analogous formulas holding for bk.

Proof. Imagine that the defining paths of integration for ak was made of flexible rubberband which is tied totk tk+1. Now movingtj will possibly drag the rubber band into somenew position. The resulting curves are precisely those paths of integration which need tobe used to compute ak. If j= k 1, k+ 1, the paths remain the same, hence ak = ak.If j = k + 1, the rubberband between tk and tk+1 is pulled around tk+2 and back totk+1. Henceak changes by the amount of the integral which goes from tk+1 totk+2, loopsaround tk+2 and then back to tk+1. Hence the first part contributesak+1. Now, for thesecond part of the path of integration, by the very definition of the Schwarz-Christoffel

maps we know that a small interval through tk+2 is mapped to a 90

hinge, so that asmall infinitesimal loop turning around tk+2 will be mapped to an infinitesimal straightline segment. In fact, locally neartk+2the Schwarz-Christoffel map is of the form zz1/2or z z 3/2. Therefore we get from the integration back to tk+1 another contribution of+ak+1. The same argument is valid for j =k 1 and for the bk.

Now denote by := tk+1tk and fix all tj other than tk+1: we regard tk+1 as theindependent variable.

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Corollary 4.4.2. The functionsak log i ak+1 and bk log i bk+1 are holomorphic in at= 0.

Proof. By Lemma 4.4.1, the above functions are singlevalued and holomorphic in a punc-tured neighborhood of= 0. It is easy to see from the explicit integrals defining ak, bk,ak

+1andbk

+1that the above functions are also bounded, hence they extend holomorphi-

cally to = 0. Now for the properness argument, we are more interested in the absolute values of the

periods than in the periods themselves: we translate the above statement about periodsinto a statement about their respective absolute values. This will lead to a crucial differencein the behavior of the extremal length functions on the NE and SW regions.

Corollary 4.4.3. Either|ak| log |ak+1| or|ak|+ log |ak+1|is real analytic infor= 0.In the first case,|bk| + log |bk+1| is real analytic in, in the second|bk| log |bk+1|.Remark. Note the different signs here! This reflects that we alternate between left andright turns in the zigzag.

Proof. If we follow the images of thetk in the NE-domain, we turn alternatingly left andright, that is, the direction ofak+1 alternates between i times the direction ofak anditimes the direction ofak.

This proves the first statement, using corollary 4.4.2. Now if we turn left at Pk in theNE domain, we turn right in the SW domain, and vice versa, because the zigzag is runthrough in the opposite orientation. This proves the second statement.

From this we deduce a certain non-analyticity which is used in the properness proof.Denote bysk, tk the preimages of the vertices Pk of a zigzag under the Schwarz-Christoffelmaps for the NW- and SW-domain respectively. We normalize these maps such thats0 =t0 = 0 and sn =tn = 1. IntroduceNE = sk+1

sk and SW = tk+1

tk. We can

now considerNE as a function ofSW:

Corollary 4.4.4. The functionNE does not depend real analytically onSW.

Proof. Suppose the opposite is true.We know that either|bk| + log SW |bk+1| or|bk| log SW

|bk+1| depends real analytically on SW, hence on NE, so we may assume withno loss in generality that|bk| + log SW |bk+1| depends real analytically on SW. Then byCorollary 4.4.3, we see that|ak| log SW |ak+1| depends analytically on NE, hence byassumption on SW. Hence B(SW) :=

|bk||bk+1|

+ log SW and A(NE) := |ak||ak+1|

log NEdepends real analytically on SW and NE, respectively. But

|bk||bk+1| =

|ak||ak+1|

by the assumption on equality of periods, so

B(SW) log SW

=A(NE) +log NE

.

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But then log(SWNE) =B(SW)A(NE) is analytic inNE. Of course, the productSWNEis analytic inNEand non-constant, asSWNEtends to zero by the hypothesis on extremallength. But then log(SWNE) is analytic inNE, nearNE= 0, which is absurd.

Remark. Note that the Corollary remains true if we consider zigzags which turn alter-natingly left and right by a (fixed) angle other than /2. This will only affect constantsin Lemma 4.4.1 and Corollaries 4.4.2, 4.4.3. In Corollary 4.4.4, we need only that thecoefficients of log are distinct, and this is also true for the zigzags with non-orthogonalsides. We will use this generalization in section 6.

4.5 An extremal length computation. Here we compute the extremal length of curvesseparating two points on the real line. This will be needed in the next section. We dothis first in a model situation: Let < 0 < 1 and consider the family of curves inthe upper half plane joining the interval (, ) with the interval (0, 1). For a detailedaccount on this, see [Oht], p. 179214. He gives the result in terms of the Jacobi ellipticfunctions from which it is straightforward to deduce the asymptotic expansions which weneed. Because it fits in the spirit of this paper, we give an informal description of what isinvolved in terms of elliptic integrals of Weierstra type.

It turns out that the extremal metric for is rather explicit and can be seen best byconsidering a slightly different problem: Consider the family of curves in S2 encirclingonly and 0 thus separating them from 1 and. Then the extremal length of is twicethe one we want.

Lemma 4.5.1. The extremal metric in this situation is given by the flat cone metric onS2 {, 0, 1, } with cone angles at each of the four vertices.Proof. Directly from the length-area method of Beurling, or see [Oht].

This metric can be constructed by taking the double cover over S2 {, 0, 1, },branched over , 0, 1, which is a torus T, which has a unique flat conformal metric(up to scaling). This metric descends as the cone metric we want to S2 {, 0, 1, }.This allows us to compute the extremal length in terms of certain elliptic period integrals.Because the covering projectionp: T S2 is given by the equation p2 =p(p 1)(p )we compute the periods ofT as

(4.2) i =

i

duu(u 1)(u )

where1 denotes a curve in and 2 a curve circling aroundand 0. We conclude that

the extremal length we are looking for is given by

Lemma 4.5.2.

(4.3) Ext() = 2 |1|2

det(1, 2)

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Proof. see [Oht] Alternatively,

(4.4) 2

1=

0du

u(u1)(u)

10 duu(u1)(u)This is as explicit as we can get.

It is evident from formulas (4.2), (4.3), and (4.4) that Ext() is real analytic on T0,4,

hence on Tsymm

0,2p+2.

Now we are interested in the asymptotic behavior of the extremal lengths Ext() as0.Lemma 4.5.3.

Ext() =O 1

log||

Proof. The asymptotic behavior of elliptic integrals is well known, but it seems worthobserving that all the information which we need is in fact contained in the geometry.For a more formal treatment, we refer to [Oht, Rain]. The period 1 is easily seen to beholomorphic in by developing the integrand into a power series and integrating term byterm; explicitly we can obtain

1() = 2

1 +

1

4 +

9

322 +

25

1283 + . . .

but all we need is the holomorphy and 1(0)= 0. Concerning 2, the general theory ofordinary differential equations with regular singular points predicts that any solution ofthe o.d.e. has the general form

= c1

log()1() + f1()

+ c21()

with some explicitly known holomorphic function f1().From a similar monodromy argument as in the above section 4.4 one can obtain that

2() i

log()1()

is holomorphic at= 0 which is simultaneously a more specific but less general statement.Nevertheless this can already be used to deduce the claim, but for the sake of completenesswe cite from [Rain] the full expansion:

2() = i

(log()1() + f1()) 4i log2

1()

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with the expansion off1() given by

f1() =n=1

(a)n(b)n(n!)2

(H(a, n) + H(b, n) 2H(1, n)) n

Here

(a)n:= a(a + 1) . . . (a + n 1)H(a, n) =

1

a+

1

a + 1+ + 1

a + n 1a= 1/2

b= 1/2

From this, one can deduce the claim in any desired degree of accuracy. Now we generalize this to 4 arbitary pointst1 < t2< t3 < t4 on the real line and to the

family of curves connecting the arc t1t2 to the arc t3t4.We denote the cross ratio of t1, t2, t3, t4 by (t1 : t2 : t3 : t4) which is chosen so that

(: : 0 : 1) = ).Corollary 4.5.4. Fort2t3, we have

Ext =O

1

log |(t1 : t2 : t3 :t4)|

Proof. This follows by applying the Mobiustransformation to theti which maps them to, , 0, 1. Remark.

Here we can already see that to establish properness we need to consider pointsti such that t2t3 while t1 and t4 stay at finite distance away.4.6 Properness of the Height Function. In this section we prove

Theorem 4.6.1. The height functionD(Z) is proper onZ, forp >2.Proof. Let Z0 be a zigzag in the boundary ofZ. We can imagine Z0 as an ordinaryzigzag where some (consecutive) vertices have coalesced. We can assume that we have acluster of coalesced points Pk = Pk+1 = . . . = Pk+l but Pk1= Pk and Pk+l= Pk+l+1(herek0 and k + lp).

We first consider the case where k 1, taking up the case k = 0 later. Denotethe family of curves connecting the segment Pk1Pk with the segment Pk+lPk+l+1 (and

their counterparts symmetric about the central point P0) in the NE domain by NE andin the SW domain by SW and their extremal lengths in NE (SW, resp.) by ExtNE(ExtSW, resp.). Now recall that the height function was defined so that

D(Z)p1j=1

exp

1

ENE(j)

exp

1

ESW(j)

2.

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where the extremal length were taken of curves encircling two consecutive points. To proveproperness it is hence sufficient to prove that at least one pair ENE(j), ESW(j) approaches0 with different rates to some order for any sequence of zigzags ZnZ0. Suppose this isnot the case. Then especially allENE(j), ESW(j) withj = k, . . . , k +lapproach zero at thesame rate for some Zn

Z0. Now conformally the pointstk, . . . , tk+l+1 are determined

by the extremal lengths ENE(j) and ESW(j) so that under the assumption, Ext NE andExtSWapproach zero with the same rate. Thus to obtain a contradiction it is sufficientto prove that

e1/ExtNE e1/Ext SW

is proper in a neighborhood ofZ0 inZ. Such a neighborhood is given by all zigzags Zwhere distances between coalescing points are sufficiently small. Especially, the quantity:=|Pk Pk+j | is small.

To estimate the extremal length, we map the NE and SW domains of a zigzag Z ina neighborhood of Z0 by the inverse Schwarz-Christoffel maps to the upper half planeand apply then the asymptotic formula of the last section, using that the asymptotics for

a symmetric pair of degenerating curve families agree with the asymptotics of a singledegenerating curve family.

Denote by NE and SW the difference tk+1 tk of the images of Pk+1 and Pk forNE and SW respectively. Because the Schwarz-Christoffel map is a homeomorphism onthe compactified domains, the quantities NE and SW will go to zero with , while thedistancestk1tk and tk+1tk+l+1are uniformly bounded away from zero in any compactcoordinate patch. Hence by Corollary 4.5.4e

1ExtNE e 1ExtSW= O 1|(tk1:tk :tk+1: tk+l+1)NE| O 1|(tk1: tk :tk+1:tk+l+1)SW|

=O( 1

NE) O( 1

SW)(4.5)

On the other hand, by corollary 4.4.4, we know that NEcannot depend analytically on SWso that one term will dominate the other and no cancellation occurs. Finally all occuringconstants are uniform in a coordinate patch and NE depends there in a uniform way on. This proves local properness near Z0 in this case and gives the desired contradiction.

In the case wherePk = 0,...,Pk+lare coalescing (herek +l < p), we use the other termsin the height function, i.e. the inequality

(4.6) D(Z)p1j=0

[ENE(j) ESW(j)]2

It is a straightforward exercise in the definition of extremal length (see lemma 4.5.1) that,sincek+l1 andk+l intersect once (geometrically), and the point Pk+l+1 converges to a

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finite point distinct from Pk,...,Pk+l, we can conclude that ENE(k+l) =O( 1

ENE(k+l1))

and ESW(k+ l) = O( 1

ESW(k+l1)). (Here we use the hypothesis that p > 2 in the final

argument to ensure the existence of a second dual curve.) Yet an examination of theargument above (see also Corollary 4.4.4, especially) shows that ENE(k+l 1) vanishesat a different rate than ESW(k+l

1), hence ENE(k+l) grows at a different rate than

ESW(k+ l). This term alone then in inequality (4.6) shows the claim.

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5. The Gradient Flow for the Height Function.5.1. To find a zigzag Z for which D(Z) = 0, we imagine flowing along the vector fieldgrad D onZ to a minimum Z0. To know that this minimum Z0 represents a reflexivezigzag (i.e., a solution to our problem), we need to establish that, at such a minimum Z0,we have D(Z) = 0. That result is the goal of this section; in the present subsection, westate the result and begin the proof.

Proposition 5.1. There existsZ0 Z withD(Z0) = 0.Proof. Our plan is to find a good initial pointZ Zand then follow the flow of grad DfromZ; our choice of initial point will guarantee that the flow will lie along a curve Y Zalong which D(Z)Y

will have a special form. Both the argument for the existence of agood initial point and the argument that the negative gradient flow on the curveYis onlycritical at a point Zwith D(Z) = 0 involve understanding how a deformation of a zigzagaffects extremal lengths on SNE and SSW, so we begin with that in subsection 5.2. Insubsection 5.3 we choose our good initial point Z, while in subsection 5.4 we check that

the negative gradient flow from Z terminates at a reflexive symmetric zigzag. This willconclude the proof of the main theorems.

5.2. The tangent space toZ. In this subsection, we compute a variational theory forzigzags appropriate for our problem. In terms of our search for minimal surfaces, we recallthat the zigzags (and the resulting Euclidean geometry on the domains NE and SW)are constructed to solve the period problem for the Weierstra data: since we are left toshow that the domains NE and SW are conformally equivalent, we need a formula forthe variation of the extremal length (conformal invariants) in terms of the periods.

More particularly, note again that what we are doing throughout this paper is relatingthe Euclidean geometry of NE (and SW, respectively) with the conformal geometry ofNE (and SW, resp.) The Euclidean geometry is designed to control the periods of theone-forms NE (and SW) and is restricted by the requirements that the boundaries ofNE (SW, resp.) have alternating left and right orthogonal turns, and that NE andSW are complementary domains of a zigzag Z in C. Of course, we are interested in theconformal geometry of these domains as that is the focus of the Main Theorem B.

In terms of a variational theory, we are interested in deformations of a zigzag throughzigzags: thus, informally, the basic moves consist of shortening or lengthening individualsides while maintaining the angles at the vertices. These moves, of course, alter the con-formal structure of the complementary domains, and we need to calculate the effect onconformal invariants (in particular, extremal length) of these alterations; those calcula-

tions involve the Teichmuller theory described in section 2.3, and form the bulk of thissubsection. We list the approach below, in steps.Step 1) We consider a self-diffeomorphism f of C which takes a given zigzag Z0 to

a new zigzag Z: this is given explicitly in5.2.1, formulas 5.1. (There will be twocases of this, which will in fact require two different types of diffeomorphisms, which welabelf and f ; they are related via a symmetry, which will later benefit us through animportant cancellation.) These diffeomorphisms will be supported in a neighborhood of a

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pair of edges; later in Step 4, we will consider the effect of contracting the support ontoincreasingly smaller neighborhoods of those pair of edges.

Step 2) Infinitesimally, this deformation of zigzag results in infinitesimal changes in theconformal structures of the complementary domains, and hence tangent vectors to the

Teichmuller spaces of these domains. As described in the opening of2.3, those tangentvectors are given by Beltrami differentials NE (and SW) on NE (and SW) and it iseasy to compute NE (and SW) in terms of

ddfz

andddf

z

. This is done explicitlyin5.2.1, immediately after the explicit computations off and f.

Step 3) We apply those formulas for NEand SWto the computation of the derivativesof extremal lengths (e.g. dd=0

ENE(k), in the notation of4). Teichmuller theory (2.3)provides that this can be accomplished through formula (2.1), which exhibits gradientvectors d ExtNE(k) as meromorphic quadratic differentials

NEk (and

SWk ) on the sphere

SNE(andSSW, resp.) As we described in2.3, Gardiner [Gar] gives a recipe for construct-ing these differentials in terms of the homotopy classes of their leaves. We describe these

differentials in5.2.2.Step 4) We have excellent control on these quadratic differentials along the (lift of the)

zigzag. Yet the formula (2.1) requires us to consider an integral over the support of theBeltrami differentials NE and SW. It is convenient to take a limit of

NE(k)NE,

and the corresponding SW integral, as the support of NE is contracted towards a singlepair of symmetric segments. We take this limit and prove that it is both finite and non-zero in5.2.3: the limit then clearly has a sign which is immediately predictable basedon which segments of the zigzag are becoming longer or shorter, and how those segmentsmeet the curve whose extremal length we are measuring. The main difficulty in taking thelimits of these integrals is in controlling the appearance of some apparent singularities: thisdifficulty vanishes once one invokes the symmetry condition to observe that the apparentsingularities cancel in pairs.

We begin our implementation of this outline with some notation. Choose a zigzagZ;letIk denote the segment ofZconnecting the pointsPk andPk+1. Our goal is to considerthe effect on the conformal geometries ofSNE and SSW of a deformation ofZ, where Ik(and Ik1, resp.) move into NE: one of the adjacent sides Ik1 and Ik+1 (Ik2 andIk, resp.) is shortened and one is lengthened, and the rest of the zigzag is unchanged.(See Figure 2.)

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I

k

Ik-1

I

k+1

I

-k-1

Figure 2

5.2.1. In this subsection, we treat steps 1 and 2 of the above outline. We begin bydefining a family of maps f which move I into NE; we presently treat the case that Ikis horizontal, deferring the vertical case until the next paragraph.

With no loss in generality, we may as well assume that Ik+1 is vertical; the moregeneral case will just follow from obvious changes in notations and signs. We considerlocal (conformal) coordinatesz = x +iycentered on the midpoint ofIk (i.e., the horizontalsegment abutting Ik at the vertex Pk ofIk nearest to the line{y = x}.) In particular,suppose that Ik is represented by the real interval [a, a], and define, forb >0 and >0

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small, a local Lipschitz deformationf : CC(5.1a)

f(x, y) =

x, + bb y

, {axa, 0yb}= R1

x, + b+b y

, {axa, by0}= R2

x, y+ + bb yy (x + + a) , {a x a, 0yb}= R3x, y +

bb yy

(x a)

, {axa + , 0yb}= R4x, y+

+ b+b yy

(x + + a)

, {a x a, by0}= R5x, y +

b+b yy

(x a)

, {axa + , by0}= R6(x, y) otherwise

where we have defined the regions R1, . . . , R6 within the definition off. Also note thathereZ0 contains the arc{(a, y) |0yb} {(x, 0) | axa} {(a, y) | by

0}.

RR R

R RR

3 1

625

a+a-a

b

-b

-a-

4

Figure 3Of course f differs from the identity only on a neighborhood ofIk; so that f(Z0) is

a zigzag but no longer a symmetric zigzag. We next modify f in a neighborhood of thereflected (across the y = x line) segment Ik1 in an analogous way with a map f

so

that f f(Z) will be a symmetric zigzag. (Here f is exactly a reflection off ifk >1.26


27/39

In the casek = 1, we require only a small adjustment for the fact that f has changed oneof the sides adjacent to the segment P1P0, both segments of which lie in supp(f

id).)

Our present conventions are that Ik is horizontal; this forces Ik1 to be vertical andwe now write down f for such a vertical segment; this is a straightforward extension ofthe description off for a horizontal side, but we present the definition off anyway, as

we are crucially interested in the signs of the terms. So set

(5.1b) f =

+ bb x, y

, {0xb, aya}= R1 + b+x , y

, {bx0, aya}= R2x +

bb xx

(y a), y

, {0xb, a ya + }= R3x +

+ bb xx

(y+ + a), y

, {0xb, a y a}= R4x +

b+b xx

(y a), y

, {bx0, aya + }= R5

x + + b+b x (y+ + a), y , {bx0, a y a}= R6(x, y) otherwise

.

Note that under the reflection across the line{y = x}, the regions R1 and R2 get takento R1 and R

2, but R4 and R6 get taken to R

3 andR

5, while R3 and R5 get taken to R

4

and R6, respectively.

Let = (f)z(fe)z

denote the Beltrami differential off, and set = dd

=0

. Similarly,

let denote the Beltrami differential off, and set

= dd=0

. Let = + . Now

is a Beltrami differential supported in a bounded domain in C = NE Z0 SWaroundZ0, so it restricts to a pair (NE, SW) of Beltrami differentials on the pair of domains

(NE, SW). Thus, this pair of Beltrami differentials lift to a pair (NE, SW) on the pair(SNE, SSW) of punctured spheres, where we have maintained the same notation for thislifted pair. But then, as a pair of Beltrami differentials on (SNE, SSW)Tsymm0,2p+2, the pair(NE, SW) represents a tangent vector toZ Tsymm0,2p+2 at Z0. It is our plan to evaluatedD(NE, SW) to a precision sufficient to show that dD(NE, SW) < 0. To do this, wecomputedExt([]) for relevant classes of curves [].

We begin by observing that it is easy to compute that =dd

=0

fz

evaluates nearIk to(5.2a)

=

12b , zR1

1

2b

, z

R2

12b [x + + a]/+ i (1 y/b) 12d = 12b (z+ + a + ib), zR3 12b [x a]/ i (1 y/b) 12 = 12b (z+ + a ib), zR4 12b [x + + a]/+ i (1 + y/b) 12 = 12b (z a + ib), zR5

12b [x a]/ i (1 + y/b) 12 = 12b (z a ib), zR60 z /supp(f id)

.

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We further compute

(5.2b)

=

12b , R11

2b , R2

12b (iz a bi) R3

12b (iz a + bi) R4

12b (iz+ + a bi) R5

12b (iz+ + a + bi) R

6

Of course, this then defines the pair (NE, SW) by restriction to the appropriate neigh-borhoods. In particular, NE is supported in the (lifts of) the regions R1,R4,R6,R

1,R

3

and R5 while SW is supported in the (lifts of) R2, R3,R5, R2,R

4 and R

6.

5.2.2. We next consider the effect of the variation dd=0

f upon the conformal geome-tries ofSNE and SSW. We compute the infinitesimal changes of some extremal lengthsinduced by the variation dd =0f.For [] a homotopy class of (a family of) simple closed curves, the form d Ext()([])T()T

symm0,2p+2 is given by an element of QD

symm(). We describe some of these quadraticdifferentials now; this is step 3 of the outline.

To begin, since the holomorphic quadratic differential NEk =NEk =d ExtSNE([k]) is

an element of QDsymm0,2p+2(SNE), it is lifted from a holomorphic quadratic differentialNEk on

NEwhose horizontal foliation has nonsingular leaves either orthogonal to and connectingthe segments Ik2 and Ik or orthogonal to and connecting the segments Ik and Ik+2.(The foliation is parallel to the other segments ofZ, and the vertices where the foliationchanges from orthogonality to parallelism lift to points where the differential NEk has asimple pole.)

Now the segmentsIkNEcorresponds under the map : NESWto the segmentIk2 SW; similarly, Ik2 NE corresponds to Ik SW. Thus d ExtSSW QDsymm0,2p+2(SSW) is lifted from a holomorphic quadratic differential

SWk whose horizontal

foliation has nonsingular leaves orthogonal to and connecting the segments Ik2 andIk and orthogonal to and connecting Ik andIk+2, in an analogous way to

NEk . Now the

foliations have characteristic local forms near the support of the divisors of the differentials,and so the foliations ofNEk (and

SWk ) determine the divisors of these differentials. We

collect this discussion, and its implications for the divisors, as

Lemma 5.2. The horizontal foliations for NEk and SWk extend to a foliation of C =

NE Z0 SW, which is singular only at the vertices ofZ. This foliation is parallel toZ except at Ik2, Ik, Ik and Ik+2, where it is orthogonal. The differential

NEk (andSWk ) have divisors

(NEk ) =P20 P

2(Pk+1PkPk1Pk2)

1(Pk1PkPk+1Pk+2)1 = (SWk ) if k >1

(NE1 ) =P2(P3P2P1P1P2P3)

1 = (SW1 )

wherePj refers to the lift ofPjZ to SNE andSSW, respectively.28


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5.2.3. Let NE denote a meromorphic quadratic differential on SNE (symmetric aboutthe lift of{y = x}) lifted from a (holomorphic) quadratic differential NE on (the opendomain) NE; suppose that

NE represents the covector d Ext([]) in TSNE

Tsymm0,2p+2 forsome class of curves []. Formula (2.1) says that

(5.3) d

d

=0

ExtSNE

([]) = 2 ReSNE

NENE= 4 Re

NE

NENE

whereSNE is the punctured sphere obtained by appropriately doubling f(NE).

The formula (5.3) is the basic variational formula that we will use to estimate thechanges in the conformal geometries of NE and SW as we vary inZ. However, in orderto evaluate these integrals to a precision sufficient to prove Proposition 5.1, we requirea lemma. As background to the lemma, note that NE and SW depend upon a choiceof small constants b > 0 and > 0 describing the size of the neighborhood of Ik andIk1 supporting NE and SW; on the other hand, a hypothesis like the foliation of

NE

is orthogonal to or parallel to Ik and Ik1 concerns the behavior of NE only at Ikand Ik1 (i.e., when b = = 0). Thus, to use this information about the foliations inevaluating the right hand sides of formula (5.3), we need to have control on Re

NE

NENEas b and tend to zero. This is step 4 of the outline we gave at the outset of section 5.2.

Lemma 5.3. limb0,0Re

NENENE exists and is finite and non-vanishing. More-

over, ifNE has foliation either orthogonal to or parallel to Ik Ik1, then the sign ofthe limit equals sgn(NENE(q)) whereqis a point on the interior ofIk Ik1.

Proof. On the interior ofIk Ik1, the coefficients of both NE

and NE have locallyconstant sign; as we see fromNE being either orthogonal or parallel toZand symmetric,and from the form of NE in (5.2a) and (5.2b). We then easily check that the sign of theproduct NENE is constant on the interior ofIk Ik1, proving the final statement ofthe lemma.

The only difficulty in seeing the existence of a finite limit as b+ 0 is the possiblepresence of simple poles ofNE at the lifts of endpoints ofIk Ik1.

To understand the singular behavior ofNE near a vertex of the zigzag, we begin byobserving that on a preimage on SNE of such a vertex, the quadratic differential has asimple pole. Now let be a local uniformizing parameter near the preimage of the vertexon SNE and a local uniformizing parameter near the vertex ofZ on C. There are twocases to consider, depending on whether the angle in NE at the vertex is (i) 3/2 or (ii)/2. In the first case, the map from NEto a lift of NEinSNEis given in coordinates by

= (i)2/3, and in the second case by = 2. Thus, in the first case we write NEk =cd2

so that NEk =4/9c(i)4/3d2, and in the second case we write NEk = 4cd2; in bothcases, the constant c is real with sign determined by the direction of the foliation.

With these expansions forNEk andSWk , we can computed Ext([k])[]; of course, this

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quantity is given by formula (2.1) as

d ExtSNE [(k])[] = 2 Re

SNE

k

= 4 ReNE k= 4 Re

R1R1

+

R4R3

+

R6R5

k.(5.4)

Clearly, as b+ 0, as|| = O max 1b , 1 , we need only concern ourselves with thecontribution to the integrals of the singularity at the vertices ofZwith angle 3/2.

To begin this analysis, recall that we have assumed that Ik is horizontal so that Z hasa vertex angle of 3/2 at Pk+1 and Pk1. It is convenient to rotate a neighborhood ofIk1 through an angle of/2 so that the support of is a reflection of the support of (see equation (5.1)) through a vertical line. If the coordinates of supp and supp

are z and z, respectively (with z(Pk+1) = z(Pk1) = 0), then the maps which liftneighborhoods ofPk+1 and Pk1, respectively, to the sphere SNE are given by

z(iz)2/3 = and z (z)2/3 =.

Now the poles on SNE have coefficients cd2

andcd2

, respectively, so we find that

when we pull back these poles from SNE to NE, we have NE(z) = 49 c dz2/2 while

NE(z) =49 c dz2/()2 in the coordinatesz and z for supp and supp , respectively.But by tracing through the conformal maps z2 on supp andz ()2,we see that ifz is the reflection ofz through a line, then

1((z))2

= 1/(z)2

so that the coefficients NE(z) and NE(z) of NE = NE(z)dz2 near Pk+1 and of

NE(z)dz2 near Pk1 satisfy NE(z) = NE(z), at least for the singular part of the

coefficient.On the other hand, we can also compute a relationship between the Beltrami coeffi-

cients (z) and (z), in the obvious notation, after we observe that f(z) =f(z).

Differentiating, we find that

(z) = f(z)z

=f(z)z= (f(z))z

= f(z)z

= (z).30


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Combining our computations ofNE(z) and (z) and using that the reflection zzreverses orientation, we find that (in the coordinates z = x +iy and z = x+iy) forsmall neighborhoodsNkappa(Pk+1) and Nkappa(Pk1) ofPk+1 andPk1 respectively,

Re supp Nkappa(Pk+1) NE(z)(z)dxdy+ Re supp Nkappa(Pk1)

NE(z)(z)dxdy

= Re

supp Nkappa(P)

NE(z)(z) NE(z)(z)dxdy

= Re

supp Nkappa

NE(z)(z) [NE(z) + O(1)] (z)dxdy

=O(b + )

the last part following from the singular coefficients summing to a purely imaginary term

while =O 1b + 1 , and the neighborhood has area b. This concludes the proof of thelemma. 5.3 A good initial point for the flow. In this subsection, we seek a symmetric zigzagZ of genus p with the property that ENE(k) = ESW(k) for k = 2, . . . , p 1. This willgreatly simplify the height function D(Z) at Z. Our argument for the existence ofZ

involves the

Assumption 5.4. There is a reflexive symmetric zigzag of genus p 1.

Since Ennepers surface can be represented by the zigzag of just the positive x- andy- axes, and we already have represented the Chen-Gackstatter surface of genus one by a

zigzag in3, the initial step of the inductive proof of this assumption is established.The effect of the assumption is that on the codimension 1 face of Z consisting of

zigzags with P1 =P0 =P1, there is a degenerate zigzag Z0 with ENE(k) =ESW(k) for

k= 2, . . . , p 1. Our goal in this subsection is the proof of

Lemma 5.5. There is a familyZt Zof non-degenerate symmetric zigzags with limitpointZ0 where each zigzagZ

t satisfiesENE(k) =ESW(k) fork= 2, . . . , p 1.

Proof. We apply the implicit function theorem to a neighborhood V of Z0 = (Z0 , 0)

in Z (, ), where we will identify a neighborhood ofZ0 inZ with a neighborhoodU ofZ0 in Z [0, ). So our argument will proceed in three steps: (i) we first defineour embedding ofU into V, (ii) we then show that the mapping : (Z, t)(ENE(2) ESW(2), . . . , E NE(p 1) ESW(p 1)) is differentiable and then (iii) finally we show thatdZ{0}

is an isomorphism ontoRp2. The first two steps are essentially formal, while

the last step involves most of the geometric background we have developed so far, and isthe key step in our approach to the gradient flow.

For our first step, normalize the zigzags in U(as in section 4.1) so that Pp = i andPp = 1 and for Z in U near Z0 , and let (t(Z), a2(Z), . . . , ap1(Z)) denote the Euclidean

31


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lengths of the segments I0, I1, . . . , I p2. Then forZU, letZhave coordinates ((Z), t)where(Z)Zhas normalized Euclidean lengths a2(Z), . . . , ap1(Z). It is easy to seethat : U Zis a continuous and well-defined map.

Next we verify that the map is differentiable. We can calculate the differential D

U

by applying some of the discussion of the previous subsection 5.2. For instance, the

matrixD U (Z) can be calculated in terms ofdENE(k) Z [] where corresponds toan infinitesimal motion of an edge ofZ, as in formula (5.3). Indeed we see that as 0,all of the derivativesd(ENE(k)ESW(k))[] are bounded and converge: this follows easilyfrom observing that the quadratic differentialsk are bounded and converge as0 andthen applying formulas (5.3) and (5.2). In fact, when supp meets the lift ofI0 I1,the same argument continues to hold, after we make one observation. We observe that wecan restrict our attention to where we are sliding only the segmentsI1, . . . , I p1 (and theirreflections) and not I0 (and I1), as the tangent space is p 1 dimensional; thus thesederivatives are bounded as well.

This is all the differentiability we need for the relevant version of the implicit functiontheorem.

Finally, we show that d Z0

: TZ0Zp1 Rp2 is an isomorphism. To see this it is

sufficient to verify that this linear map dZ0

has no kernel. So letvTZ0Zp1, so that

v=

p1i=1

cii

where ci R and i refers to an infinitesimal perturbation ofIi and Ii1 into NE (inthe notation for zigzags inZp: forZ0 , we have that I0 and I1 have collapsed onto P0.

Suppose, up to looking atv instead ofv , that someci > 0, and let{ij}be the subsetof the index set

{1, . . . , p

1

} for which cij > 0. We consider the (non-empty) curve

system of arcs connecting Iij to the intervalPpP andIij1 toPPp, let v denotea Jenkins-Strebel differential associated to this curve system. By construction, sgn v isconstant on the interior of every interval, and v > 0 on the interior ofIi if and only ifthe index i {ij}.

Thus, by Lemma 5.3 and formula (5.2), we see that both

(5.5a) d Ext(NE)[v] =

p1i=1

ci

Ii

vi>0

and from formulas (5.2) and (5.3), using that the horizontal (and vertical) foliation(s) ofv extend to SW, that

(5.5b) d Ext(SW)[v] =

p1i=1

ci

Ii

vi 0.32


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Now suppose that dZ0

[v] = 0. Then by the definition of , we would have that

d Ext(NE)[v] = d Ext(SW)[v], and sinceExti(NE) provides local coordinates inTeichmuller space, we would see that the Teichmuller distance between NE and SWwould infinitesimally vanish. But that would force ExtNE() ExtSW() to vanish tofirst order, which contradicts our computation (5.5).

We conclude thatd Z0 is an isomorphism, so that the implicit function theorem yieldsthe statement of the lemma.

5.4 The flow from the good initial point and the proofs of the main results. Wenow consider one of our good zigzags Zt Zp and use it as an initial point from whichto flow along grad D(Z) to a reflexive symmetric zigzag.

LetY Zp denote the set of genus p zigzags for which ExtNE(i) = ExtSW(i) fori= 2, . . . , p1. As extremal length functions are inC1(Tsymm0,2p+2) by Gardiner-Masur [GM],we see thatYis a piecewiseC1 submanifold ofZp. (We shall note momentarily that in ourcase, these extremal length functions are real analytic.) We consider the height function

D restricted to the setY.

Lemma 5.6. DY

is proper and is critical only at pointsZ Y Zfor whichD(Z) = 0.

Proof. The properness of DY

follows from the properness of D, as shown in4. Wenext show that ifD(Z)= 0 for Z Y, then there exists a tangent vector v TZZ forwhichdD[v]< 0, and for whichd (ExtNE(i) ExtSW(i)) [v] = 0 so that v lies tangentto a fragment ofY and infinitesimally reduces the height D.

Indeed, we observe that

DY

=

exp 1

ENE(1)

exp 1

ESW(1)

2+

ENE(1) ESW(1)2

as the other terms vanish.

Now, observe that Zis a real analytic submanifold of the real analytic product manifoldTsymm0,2p+2 Tsymm0,2p+2, being defined in terms of periods of a pair of holomorphic forms on theunderlying punctured spheres. Next we observe thatENE(j) and ESW(j) are, for each j,real analytic functions on Tsymm0,2p+2 with non-degenerate level sets. To see this note thatthe extremal length functions correspond to just the energy of harmonic maps from the

punctured spheres to an interval, with the required analyticity coming from Eells-Lemaire[EL], or directly from 4.5; the non-degeneracy follows from Lemma 5.3, if we apply anyof the form (5.2) to the zigzag (this will be developed in more detail in the followingparagraph). Thus, the setYacquires the structure of a real analytic submanifold properlyembedded in Z. AsYis one-dimensional nearZ0, it is one-dimensional (with no boundarypoints) everywhere.

Now, for Z Y Z which is nota zero ofD, we have for any tangent vector the33


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formula

dD[] = 2

exp

1

ExtSNE([1])

exp

1

ExtSSW([1])

exp 1ExtSNE ([1]) (ExtSNE([1]))2 d ExtSNE([1])[]+ exp

1

ExtSSW([1])

(ExtSSW([1]))

2d ExtSSW([1])[]

+ [ExtSNE([1]) ExtSSW([1])](d ExtSNE([1])[] d ExtSSW([1])[])

Then if we evaluate this expression for, say, being given by lifting an infinitesimal moveof just one side as in formulas (5.1) and (5.2), we find by an argument similar to that forinequalities (5.5) and (5.6) that dD[]= 0. This concludes the proof of the lemma.

Conclusion of the proof of Proposition 5.1. We argue by induction. The union

of the positive x- and y-axes, is reflexive via the explicit map z iz3

; this verifies thestatement of the proposition for genus p = 0. There is also a unique reflexive zigzag forgenusp = 1, after we make use of the permissible normalization P1= 1; we can verify thatbothSNE andSSWare square tori, as in the first paragraph of3. In general, once we aregiven a reflexive symmetric zigzag of genus p 1, we are able to satisfy Assumption 5.4,so that Lemmas 5.5 and 5.6 guarantee the existence of a one-dimensional submanifoldY Zalong which the height function is proper. A minimum Z0 for this height functionis critical forD alongY, and hence satisfies D(Z0) = 0 by Lemma 5.6.

As the discussion of subsection 4.3 shows that ifD(Z0) = 0, then Z0 is reflexive, weconclude from Proposition 5.1 the

Main Theorem B. There exists a reflexive symmetric zigzag of genusp forp

0 whichis isolated inZp.Proof of Main Theorem B. The local uniqueness follows from inequality (5.6) and theargument following it.

Our main goal then follows.

Proof of Main Theorem A. By Main Theorem B, there exists a symmetric reflexivezigzag of genusp. By Theorem 3.3, and Lemma 3.4 from this zigzag we can find Weierstradata for the required minimal surface.

5.5 A remark on a different height function. In this subsection, we try to give somecontext to the methods we adopted in Sections 4 and 5 by considering an alternate and

perhaps more natural height function.Define a new height function

H(Z) ={dTeich(RNE, RSW}2.

CertainlyH(Z) = 0 if and only ifZ is reflexive. Moreover, here the gradient flow is mucheasier to work with, at least locally inZp. Indeed, we observe

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Lemma 5.7. H(Z) = 0 if and only ifZis critical forH() onZp.Sketch of Proof. Clearly H(Z) 0 and H is C1 (even real analytic by the proof ofLemma 5.5) onZpTsymm0,2p+2 Tsymm0,2p+2, so ifH(Z) = 0, then Zis critical for H onZp.

So suppose that H(Z)= 0. ThenRNE is not conformally equivalent toRSW, so wecan look at the unique Teichmuller map in the homotopy class [: S

NES

SW] (see the

opening of3).From the construction ofSNE, SSW and from NE, SW and , we can draw many

conclusions about the Teichmuller differentialsqQDsymm(SNE) andq QDsymm(SSW).For instance, the foliations are either perpendicular or parallel to the image ofZ on SNEandSSW, and by the construction of the Teichmuller maps, there are zeros in the lift of anintervalIk on SNE if and only if there is a corresponding zero in the lift of(Ik) on SSW.Moreover, there is a simple pole at a puncture on SNE if and only if there is a simple poleat a corresponding puncture on SSW.

Finally, we observe that qhas only simple poles and as the foliation ofq is symmetricabout both the images ofZand the line{y = x}, we see that there cannot be a simplepole at either the lift ofP0 or. Yet by Riemann-Roch, as there are 4 more poles thanzeros, counting multiplicity, there must be a pair of intervals Ik Ik1 whose lift has nozeros ofq(and whose image under of lift has no zeros ofq QDsymm(SSW)).

Observe next that Kerckhoffs formula (2.2) says that the horizontal measured foliation(Fq, q) ofq extremizes the quotient on the right hand side of 2.2. However, consider adeformation (5.1) on our zero-free intervals Ik Ik1. By Lemma 5.3 and formula (5.3)we see that either

d ExtSNE(q)[]>0 and d ExtSSW(q)[]< 0

or

d ExtSNE(q)[]< 0 and d ExtSSW(q)[]>0.

In either case, we see from Kerckhoffs formula that dHZ

[]= 0, and so Z is notcritical.

So why use our more complicated height function? The answer lies in formula (4.2),which combined with Kerckhoffs formula (2.1) shows that H() :ZpR is not a properfunction onZp. Thus, the backwards gradient flow might flow to a reflexive symmetriczigzag, or it may flow to Zp. A better understanding of this height function H onZwould be interesting both in its own right and if it would lead to a new numerical algorithmfor finding minimal surfaces experimentally.

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6. Extensions of the Method: The Karcher-Thayer Surfaces.Thayer [T], following work of Karcher [K], defined Weierstra data (depending upon

unknown constants) for a family of surfaces Mp,k of genus p(k1) with one Enneper-type end of winding order 2k 1. In this notation, the surfaces of genus p described inTheorem A are written Mp,2. Karcher [K] has solved the period problem for the surfaces

M0,k and M1,k for k > 2 and Thayer has solved the period problem for M2,k for k > 2.He has also found numerical evidence supporting the solvability of the period problem for

p34, k9. Here we proveTheorem C. There exists a minimally immersed surfaceMp,k of genusp(k 1) with oneEnneper-type end with winding order2k 1.Proof. We argue in close analogy with our proof of Theorem A (k= 2). LetZp,k denotethe space of equivalence classes of zigzags with 2p+ 1 finite vertices and angles at thevertices alternating between /kand 2k1k . We double the complementary domains

kNE

and kSW of Zto obtain cone-metric spheres with cone angles of alternating 2/k and2/k(2k

1). We then take a k-fold cover of those spheres, branched at the images of the

vertices of the zigzag on the cone-metric spheres, to obtain Riemann surfacesRkNE andRkSW. By pulling back the form dz from kNE and kSW, we obtain, as in Section 3, a pairof forms = gdhand =g1dhon which we can base our Weierstra representation. As

before, we set dh= d , where is the branched covering map :RkNEC, so that dhis exact; as before, the periods of andare constructed to be conjugate, as soon as thezigzag is reflexive, and the induced metric (2.0) is regular at the lifts of the finite verticesof the zigzag.

To see that we can find a reflexive zigzag withinZp,k, we observe that by the remark atthe end of section 4.4, the same real non-analyticity arguments of Section 4.4 carry overto the present case, once we replace the /2 and 3/2 angles with /k and /k(2k 1)angles. All of the rest of the arguments of Section 4 carry over without change and weconclude that the height function D(Z) is proper onZp,k.

For the gradient flow, we can write down deformations along the zigzag analogous toformula (5.1) (it is sufficient just to conjugate the maps in 5.1 by a map which shears theoriginal zigzag with vertex angles /2 and 3/2 to a zigzag with angles /k and 2k1k )and then check that the proof of Lemma 5.3 continues to hold. The rest of the argumentsin Section 5 carry over unchanged to the present case. Thus we find a reflexive symmetriczigzag inZp,k via the proof of Theorem B. The present theorem then follows. Remark. Of course, the arguments in the last two paragraphs of the proof of TheoremC apply equally well to zigzags of arbitrary alternating angles and 2 , so one mightwell ask why we do not generalize the statement of Theorem C even farther. The answerlies in that while the Teichmuller theory of sections 4 and 5 extends to zigzags with non-orthogonal angles, the discussion in section 3 of the transition from the zigzags to regularminimal surfaces only extends to the zigzags described in the proof of Theorem C. Forinstance, we of course require a finitely branched cover over a double of the zigzag in orderto get a surface of finite genus, so we must restrict our attention at the outset to zigzagswith rational angles. However, if the smaller angle should be of the form = rs, we

36


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find that an s-sheeted cover of the double of a zigzag would be forced to have an inducedmetric (2.0) which was not regular at the lifts of the finite vertices.

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