Infinitesimal CR automorphisms of rigid hypersurfaces in C2

The Journal of Geometric Analysis Volume 1, Number 3, 1991

Infinitesimal CR Automorphisms Hypersurfaces in C 2

By Nancy K. Stanton

of Rigid

ABSTRACT. In this paper, we describe the space of infinitesimal CR automor-

phisms of a rigid, real analytic, real hypersurface in C 2. We use these results to obtain a geometric characterization of the homogeneous hypersurfaces. Here, a hypersurface is called homogeneous if it is equivalent to one given by an equation of the form I ra(w) = p where p is a homogeneous polynomial in z and 3. This gives an answer in dimension 2 to a problem posed by Linda Rothschild. We give another answer, in terms of a normal form for the defining function, in our paper "A normal form for rigid hypersurfaces in C 2.''

Let M be a real hypersurface through the origin in C n+l, or, more generally, an integrable CR manifold of hypersurface type. A smooth vector field X on M is called an infinitesimal CR automorphism of M if the local one-parameter group it generates is a local group of CR automorphisms of M. In this paper we consider the following local problem:

Suppose M is a rigid, real analytic, real hypersurface through the origin in C 2. What are

the infinitesimal CR automorphisms of M that are defined in a neighborhood of 0?

Here, following the terminology of Baouendi, Rothschild, and Treves [BRT], we call M rigid if there are coordinates (Z l , . . . , zn, w) such that M is given by an equation of the form

Imw = F(z,-Z). (0.1)

This is equivalent to the existence of a transversal infinitesimal CR automorphism on M. Rigid hypersurfaces are called regular by Tanaka [T] and T-regular by D'Angelo [D]. If M is locally

CR isomorphic to the plane

Imw = 0, (0.2)

Math Subject Classification: 32F25. Key Words and Phrases Infinitesimal CR automorphisms, real hypersurfaces, rigid hypersurfaces. Research supported in part by NSF Grants DMS 86--01267, DMS 89-01547, and RII 86--00042.

Q1991 CRC Press, Inc ISSN 1050-6926

232 Nancy K. Stanton

the space V of infinitesimal CR automorphisms is infinite dimensional. In C 2, if M is rigid, real analytic, and not locally CR isomorphic to (0.2), it is of finite type and V is finite dimensional.

If M is rigid, dim V _> 1. However, rigidity is not necessary for the existence of infinitesimal CR automorphisms. A simple example of a hypersurface that is not rigid but has nontrivial infinitesimal CR automorphisms is the hypersurface M given by the equation Im w = Re wN 2. The origin is a point of infinite type on M, but M is not Levi flat so it is not equivalent to (0.2). Hence it is not rigid. Because M is invariant under rotations in the variable z, the vector field Re izO/Oz is an infinitesimal CR automorphism of M.

Global automorphism groups of three-dimensional CR manifolds were studied by E. Car- tan [C]. He gave a complete list of the three-dimensional CR manifolds that admit a transitive automorphism group. Such manifolds are necessarily strongly pseudoconvex. He did not classify the three-dimensional CR manifolds that admit a nontrivial nontransitive automorphism group. Although he stated the local problem, he did not analyze it. The group of CR automorphisms fixing a point of a nondegenerate hypersurface has been analyzed by Belogapka [B], Ezhov, Kruzhilin, Loboda, and Vitushkin. In [K], Kruzhilin gives a survey of these results.

We give the background in Section 1. Our starting point is to assume that the hypersurface is in rigid normal form. This normal form was introduced in [S1]. In Section 2, we characterize infinitesimal CR automorphisms. A corollary is that the space of such automorphisms is a Lie algebra. In Theorem 3.3 we describe the space of infinitesimal CR automorphisms in the case that M is given by an equation of the form

Imw=p(z,2) (0.3)

where p is a homogeneous polynomial. In Section 4 we introduce the weight of a vector field. The weight gives a filtration of the space of infinitesimal CR automorphisms of M. The lowest weight part of an infinitesimal CR automorphism of M is an infinitesimal CR automorphism of the homogeneous part of M. We show in Theorem 4.23 that the lowest weight part of an infinitesimal CR automorphism sometimes determines M up to CR equivalence. After giving some general results on the form of infinitesimal CR automorphisms in Section 5, we prove Theorem 3.3 in Section 6. In Section 7, we determine the infinitesimal CR automorphisms of M if the homogeneous part of M is not equivalent to a tube. In Section 8 we discuss the case that the homogeneous part of M is equivalent to a tube and M is not strongly pseudoconvex. We analyze the strongly pseudoconvex case in Section 9. The proofs use power series techniques, ordinary differential equations, and the Lie algebra structure of the space of infinitesimal CR automorphisms.

A rigid hypersurface is called homogeneous if it is locally equivalent to (0.3) with p a homogeneous polynomial. This terminology comes from the fact that an equation of the form (0.3) with p a homogeneous polynomial is homogeneous with respect to a nonisotropic group of dilations (see Section 4). In Theorem 4.23 we characterize the homogeneous hypersurfaces given by a defining equation in rigid normal form. We characterize the rotationally invariant hypersurfaces and the homogeneous hypersurfaces with equations of a more general form in Section 10. In particular, in Theorem 4.23 and Theorem 10.14 we obtain an answer in dimension 2

Infinitesimal CR Automorphisms of Rigid Hypersurfaces in C 2

to a question posed by Linda Rothschild:

233

How can you tell if a rigid hypersurface is homogeneous?

We apply the results of this paper in [S 1 ] to study the question of when two hypersurfaces in rigid normal form are CR equivalent. This allows us to answer Rothschild's question in terms

of the rigid normal form IS 1, Theorem 7.1 ].

I want to thank Linda Rothschild and Salah Baouendi for helpful discussions concerning this

work. Some of the work for this paper was done while I was visiting the University of Michigan.

I wish to thank the Mathematics Department at the University of Michigan for its hospitality. I also want to thank the referee for very helpful comments.

1. Background

Throughout this paper, unless we state otherwise, we will assume that M is a rigid, real analytic, real hypersurface of finite type through the origin in C 2. We will work locally; all functions, maps, vector fields, etc., will be defined in sufficiently small neighborhoods of the

origin and all maps will preserve the origin. One of the main results of [S1] is that there is a

complete rigid normal form for the equation of M .

T h e o r e m 1.1 [S1, T h e o r e m 1.7]. There are coordinates (z, w = u q- iv) such that in these coordinates M is in (rigid) normal form, i.e., M is given by an equation of the form

V = Z ai j z i ~ J .qt_ i q - j : , , ~ I <_i<io i , j~_ i 0

(1.2)

where m >_ 2 and io >_ 1 are integers, the hi are holomorphic functions, hi = O ( m - i + 1), b is a real analytic function, b = O ( r a - 2io - 1), and aio,m_i o = 1. In addition, if the right side of(1.2) is a function of Izl 2, the coefficient of Izl 4i0 is o.

Here, we write f = O ( j ) to mean that f (z , -2) vanishes to order j at the origin.

Let p denote the first sum on the right side in (1,2). The hypersurface v ---- p(z,-g) is called the homogeneous part of M. The integer m is called the degree of the homogeneous part.

We say M is (locally) equivalent to a tube if it is CR isomorphic to a hypersurface given

by an equation of the form

v = h(z + (13)

The space of infinitesimal CR automorphisms of M is most complicated to analyze when the


homogeneous part has the form

m - 1

V ~ - - 03 3 - 1

m j

where w is an (m - 2)nd root of unity. We make an additional normalization in this case which makes it easy to see that (1.4) is equivalent to a tube.

Proposi t ion 1.5. Let M be the homogeneous hypersurface given by (1.4). There is a

change of coordinates that preserves normal form such that, in the new coordinates, M is given by

m j=l J

Hence, M is equivalent to

v ' = • + z--7) m, (1.7)

and to

v " = • '' - (1.8)

which are tubes.

Proof. Make the change of variables

Z : 0,3--1/2Z t, W : w - - l q - m / 2 w t . (1.9)

This takes (1.4) to (1.6) and preserves normal form. If M is given by (1.6), the change of variables

z' =- z, w' ~- • + 2iz TM) (1.10)

takes it to (1.7) and the change

takes (1.7) to (1.8). [ ]

z" = - i z ~, w" = w ~ (1.11)

R e m a r k 1.12. In the strictly pseudoconvex case, ra = 2, the result of Proposition 1.5 is particularly easy to see. Expanding the equation of the tube

v' ---- (z' + ~)2 (1.13)

Infinitesimal CR Automorphisms of Rigid Hypersurfaces in C 2 235

gives v' = z '2 + 21z'12 + ) -72. Hence the tube (1.13) is equivalent to the hyperquadric v = Izl 2 via the change of variables

Similarly, the tube

is equivalent to the hyperquadric via

z ' = z~ w ' = 2 w - k 2 i z 2. (1.14)

v' = (z' -- ~)2 (1.15)

z' = z, w' = - ( 2 w - 2i22). [ ] (1.16)

Remark 1.17. Suppose M is given by an equation of the form

v ' : F(z', (1.18)

where F vanishes to second order at the origin. Then there is a change of variables

z = az ' + f ( z ' ) , w = + w ' + 9 ( z ' ) , (1.19)

where f , 9 = O(2), taking m to an equation of the form (1.2) with hi = O ( m - i + 1), b = O ( m - 2i0 + 1). This follows from [S1, proof of Theorem 1.7]. If the right side of (1.2) is a function of Izl 2, the normalization of the coefficient of Izl 4i~ in Theorem 1.1 is given by a map of the form

z = z ' e i~ w = w ' . [] (1.20)

2. A characterization of infinitesimal CR automorphisms

In this section, we assume that M is a real analytic, real hypersurface of finite type in C e.

Definition 2.1. A smooth vector field on M is called an infinitesimal CR automorphism of M if the local one-parameter group it generates is a local group of CR automorphisms of M .

We have the following characterization of infinitesimal CR automorphisms.

Theorem 2.2. A smooth vector field X on M is an infinitesimal CR automorphism if and

only if there is a holomorphic vector field Z defined in a neighborhood of M whose real part is

equal to X on M .


By a holomorphic vector field we mean a vector field of type (1,0) whose components (with respect to the coordinate vector fields) are holomorphic functions, or, equivalently, a vector field that takes holomorphic functions to holomorphic functions.

Proof . Let X be an infinitesimal CR automorphism of M and let qSt be the local one- parameter group generated by X. Then for each t, qSt is a CR diffeomorphism from its domain Dt onto its range. By [BJT, Theorem 7.3], qSt is analytic and therefore extends to a biholomorphic

map ~t defined in a neighborhood/gt of Dr. It suffices to assume that 0 E M and to show that the theorem is true in a neighborhood of 0. Let (z, w = u + iv) be coordinates in C 2 such that M is given in a neighborhood of 0 by v = F(z,-s u). We use (z, ~, u) as local coordinates on M. By [MZ, Section 5.2], there is a neighborhood V of 0 in M and there is e > 0 such that for It[ < e the power series expansion about 0 (in (z, ~, u)) of ~bt converges in V. Hence there is a

neighborhood 0 C U in C 2 such that ~t is holomorphic in U for It] < e. Also,

(2.3)

so, because the functions in (2.3) are holomorphic, (2.3) holds on all of Dom(q~t o ~bs). Hence ~t is a local one-paranaeter group of holomorphic transformations defined in a neighborhood U of 0. By [MZ, Section 5.2], this local one-parameter group is differentiable in t at t ~- 0. Let )~ be the vector field induced by qSt on U. Then X is the real part of a holomorphic vector field [KN, Proposition IX.2.11 ].

Conversely, suppose that X is a smooth vector field on M and that Z is a holomorphic vector field defined in a neighborhood of M with Re Z = X on M. Then by the remarks preceding [KN, Proposition IX.2.10], the local one-parameter group of Re Z is a local group of biholomorphic transformations. Thus, the transformations in the local one-parameter group of X are the restrictions to M of holomorphic maps, hence CR automorphisms. [ ]

Coro l l a ry 2.4. The space V of infinitesimal CR automorphisms of M is a Lie algebra.

Proof . Let X~ Y C V. By Theorem 2.2, there are holomorphic vector fields Z and W defined in a neighborhood of M such that X ---- Z q- Z and Y = W q- W. It follows that [X, Y] -- [Z, W] _jr [Z, W]. Hence, by Theorem 2.2, IX, Y] E V. [ ]

R e m a r k 2.5. We use the hypothesis that M is of finite type in C 2 in the proof of Theorem 2.2 only to conclude that each qSt is analytic. Hence, Theorem 2.2 and Corollary 2.4 remain true if we assume that M is a real analytic hypersurface in C n+l with the property that every CR diffeomorphism on M is real analytic. In particular, by [BJT], this property holds if M is essentially finite. [ ]

Infinitesimal CR Automorphisms of Rigid Hyperswfaces in C 2 237

We will use the characterization of infinitesimal CR automorphisms given by Theorem 2.2 throughout the rest of this paper.

3. Th e homogeneous case

We say the hypersurface M is homogeneous if it is equivalent to

v = p ( z , 2 ) (3.1)

where p is a homogeneous polynomial. If M is homogeneous, by Theorem 1.1 and Proposition 1.5, we may assume that M is given by an equation of the form

V = E ai j z i~ j (3.2) i+j=m i , j_>~ o

with m > 2, io > 1, aio . . . . io = 1, and if M is equivalent to a tube, (3.2) is of the form (1.6).

Homogeneous hypersurfaces are the simplest examples of rigid hypersurfaces of finite type. If m ---- 2, the hypersurface M is just the (strictly pseudoconvex) hyperquadric, v ---- Izl 2, and the space of infinitesimal CR automorphisms of M form is naturally isomorphic to the Lie algebra su(2, 1) [C]. The following theorem gives a complete description of the space of infinitesimal CR automorphisms for any homogeneous hypersurface of finite type.

T h e o r e m 3.3. Let M be a homogeneous hypersurface of finite type in C 2 given by (3.2), and, if M is equivalent to a tube, by (1.6). Let X be an infinitesimal CR automorphism of M. Then there are holomorphic functions f and g defined in a neighborhood of 0 such that

X = 2Re ( f O + g ff-ff--~ ) m (3.4)

and there are constants a, e E R, b, c, d E C such that f and g are given by

f = b + c z + d w + 2 i d z 2 + e z w

9 = a + 2ibz m-1 + ( R e c ) m w + 2iclzw + 17~eW2. 2

(3.5)

In addition, c E R if m yL 2i0, d = 0 if m ~ 2, e = 0 if rn ~ 2/0, b = 0 if M is not equivalent to a tube and -b = - b if m > 3.

Some of the parameters in (3.5) have very simple geometric interpretations. The parameter a generates translations along the u axis. If M is equivalent to a tube, the parameter b E i R corresponds to the generator of translations of the tube (1.7) along the y axis (where z = x + iy). The parameter Re c generates dilations. The parameter Im c generates rotations in z if M is given by v = Iz] 21~


We prove Theorem 3.3 in Section 6. In the next section, we will see in Proposition 4.10

that, in addition to giving all the infinitesimal CR automorphisms in the homogeneous case, Theorem 3.3 provides some information in the nonhomogeneous case.

4. Weights and the Lie algebra structure

By Corollary 2.4, the space of infinitesimal CR automorphisms of M is a Lie algebra. This Lie algebra has a natural filtration. In this section, we use the filtration to show that an infinitesimal

CR automorphism of M induces one on the homogeneous part of M . In Theorem 4.23, we use this induced infinitesimal automorphism to characterize homogeneous hypersurfaces.

To describe the filtration, we first introduce some notation. Let M be a rigid hypersurface

in normal form, and suppose that the degree of its homogeneous part is m. Define nonisotropic dilations on C 2 by

(4.1)

for t > 0. Then tSt is an automorphism of the homogeneous part of M. We say a function h is homogeneous of weight j if h o t5 t = tJh. We say h = O(j) if every term in its Taylor expansion about 0 has weight > j . If h = h(z, ~), then h = O(j) if and only if h vanishes to

order j at 0, so this notation is consistent with the notation in Theorem 1.1.

Definition 4.2. A vector field Y is homogeneous of weight j if

Y ( f o 6t) = t - ~ ( Y f ) o 6t (4.3)

where tSt denotes the dilation (4.1).

Thus, a vector field Y is homogeneous of weight j if and only if it takes functions that are homogeneous of weight k to functions that are homogeneous of weight j + k. In particular, the vector field O/Ow is homogeneous of weight - m and the vector field O/Oz is homogeneous of weight - 1 . We write Y = O(j) if, when we expand the coefficients of Y in a Taylor series

about 0, each term is a vector field of weight > j . If Y -- O(j) and f is a function with

f = O(k), it follows that Y f -- O(j + k).

Definition 4.4. A vector field X has weight j if X = O(j) and X • O(j + 1). Let w ( X ) denote the weight of X .

Proposition 4.5. Let X, Y E V. If X = O ( j ) and Y = O ( k ) then [X, Y] = O ( j + k )


Proof. This is a straightforward calculation, using the fact that differentiation in z lowers homogeneity by 1 and differentiation in w lowers homogeneity by m. []

Let V be the space of infinitesimal CR automorphisms of M. Let Wj = { X C V : X = O(j)}. Then we have a natural filtration

. . . W j c w j _ l c . . . c w _ m = V. (4.6)

By Proposition 4.5, [Wj, Wk] C Wj+k.

Let M0 be the homogeneous part of M and let Vo be the space of infinitesimal CR automorphisms of Mo. By Theorem 3.3, the space V0 has a natural grading

V = + Vo,j, j = - m

Vo,j = {X E Vo: X is homogeneous of weight j} (4.7)

and [Vo,j, Vo,k] C Vo,j+k. Also by Theorem 3.3, 2 _< dim Vo _< 8 since O/Ou C Vo-,~ and 2Re (z(O/Oz) + mw(O/Ow)) E Vo,o.

Let X be an infinitesimal CR automorphism of M with w ( X ) = n. By Theorem 2.2, there are holomorphic functions f and 9 such that X is given by

X = 2 R e ( f - ~ z q - 9 ~ - ~ ) M" (4.8)

Write

f : E f~" 9 = E 9~,, (4.9) u>n+l v>_n+m

where f~ and 9~ are polynomials homogeneous of weight u.

Proposition 4.10. and let X be the infinitesimal CR automorphism (4.8) of M. Then

Xo = 2Re fn+ l~z z

is an infinitesimal CR automorphism of the homogeneous part Mo of M and Xo C Vo,n.

Let M be a rigid, real analytic hypersutface in normal form in C 2,

(4.11)

Proof. We may assume that the equation of M is given by

v = (4.12)

240 Nancy K, Stanton

where F is the right side of (1.2). Because X is tangent to M,

1 2~(g(z , u + iF(z ,~ ) ) - ~(~, u - iF(z ,~) ) )

�9 O F ( z , ~ ) . _ O F ( z , ~ )

= f ( z , u + ~F(z, ~)) Oz + ](~' u - ,F ( z , z)) ~ z (4.13)

Equating terms of lowest weight in (4.13) gives

L ( g n + m (Z,U -4- i p ( z , z ) ) -- gnWm (-Z,U-- ip(z,'-Z))) 2i

Op( z, -f ) Oz

- - ( 4 . 1 4 )

where p denotes the terms in F that are homogeneous of degree m. Equation (4.14) says that Xo is an infinitesimal automorphism of Mo. Because Xo is homogeneous of weight n,

Xo E Vo,n. []

Proposition 4.10 says that, if we write X = X0 + Y where X0 is homogeneous of weight n and Y z O(n + 1), then X0, the lowest weight part of X, is an infinitesimal CR automorphism of Mo.

Corol la ry 4.15. Let M be a rigid, real analytic hypersurface in normal form in C 2 and let V be the space of infinitesimal CR automorphisms of M. Let Mo be the homogeneous part of M and let Vo be the space of infinitesimal CR automorphisms of Mo. Then

dim V < dim Vo. (4.16)

and Proof. By Proposition 4.10 and Theorem 3.3, Wj = 0 for j > m. Hence, d imV < cx~

V ~ W -= ~ Wj /Wj+I . (4.17) j=-m

For each j , define a linear map

: WjlW +l Vo, (4.18)

by X ~ X0, where ]~" is a representative of X in Wj. Then Tj is injective. Let T be the linear map from W to V0 whose restriction to Wj/Wj+I is given by Tj. It follows that T is injective, so dim W < dim V0. The corollary now follows from (4.17). [ ]

We know that the vector field

]g0=2Re z ~z + r o w (4.19)


is an infinitesimal CR automorphism of M0. This vector field has the nice property that if X is a homogeneous vector field of weight j , then

In particular, setting

[go, X] = i X . (4.20)

0 Y-m = 2Re - -

Ow '

we have [go, Y_~] = - r a Y - r e . Also, if X is not homogeneous,

[g0, X] = w ( X ) X + 0 (w(X) q- 1).

(4.21)

(4.22)

The next theorem shows that the existence of an infinitesimal CR automorphism with lowest weight part Y0 characterizes homogeneous hypersurfaces. This gives an answer to Rothschild's question in C 2.

T h e o r e m 4.23. Let M be a rigid, real analytic hypersurface in normal form in C 2, and suppose M has an infinitesimal CR automorphism of the form X = Y0 + O(1). Then M is homogeneous. Conversely, if M is a homogeneous hypersurface in rigid normal form, then M has an infinitesimal CR automorphism X = go + O(1).

We will see in Section 10 that the hypothesis on the form of the equation for M can be weakened. Before we prove the theorem, we prove two propositions. We state the propositions in greater generality than is needed for the proof of Theorem 4.23. The added generality does not affect the proofs, and we need the generality in some of the proofs in Section 9.

Propos i t ion 4.24. Let M be in rigid normal form. Suppose M has infinitesimal CR automorphisms X = 2Re (c~z(O/Oz)+ m(Reee)w(O/Ow)) + O(1), c~ 5~ 0 C C, and X - m = Y-m + O( - -m + 1) such that [X, X_,~] = - r a R e ctX_,~. Then there is a biholomorphic change of coordinates such that, in the new coordinates (z ~, w~),

(i) X_m = 2Re (O/Ow');

(ii) M is given by an equation of the form v' = F'(z ' , U) where F' has no pure terms;

(iii) X = 2Re ((c~z' + f ( z ' ) ) (O/Oz ' ) + ra(Rec~)w'(O/Ow')) with f = 0 (2 ) .

Proof . By Theorem 2.2, X - m = 2Re Z_,,~ for some holomorphic vector field Z_m with

Z_,~ 10 ~ 0. Thus, we can find a biholomorphic change of coordinates preserving the origin such that in the new coordinates (z", w") , Z_m = O/Ow". Because Im Z_m is not tangent to M and Re Z_m is, in the new coordinates M has an equation of the form v" = F"(z", z"). A


change of variables of the form z' = z ' , w' = w" + h(z") for some holomorphic function h gives

0 Z - m - Ow' (4.25)

and brings the defining equation for M to one of the form

v' = F'(z ' , ~ ) (4.26)

where F' has no pure terms. Now, X is the real part of a holomorphic vector field 2Z and [Z, Z_m] -- -m(Rece)Z_m. Hence,

0 0 (4.27) Z = f'(Z')Oz~ + (m(Rec~)w' + 9'(z'))Ow '

where f ' and g' are holomorphic and vanish at 0. Applying twice the real part of (4.27) to (4.26) gives an equation of the form Imy ' (z ' ) = G ' ( z ' , z 7) where G' has no pure terms. Thus Im g' ~ 0. Because g' is holomorphic and vanishes at 0, we conclude that 9' = 0. Hence, the lowest weight part of Z is

, 0 Zo = flz ~z ~ + re(Re o~)w' O0---w ' (4.28)

for some/3 E C. Write (z', w') = A(z, w)+ higher order terms where A is linear. By (4.28) and (4.25), A(z, w) = (az + bw, w), so, by (4.27),/3 = c~. Thus

0 0 Z = z' (ol + z'qS(z')) Oz---- 7 + re(Re ct)w' (4.29)

Ow'

for some holomorphic function r [ ]

Proposition 4.30. Let Z1 = (az + f (z ) ) (O/Oz) for some a ~ 0 E C and some holomorphic function f with f = 0(2) . Then there is a holomorphic change of coordinates in C preserving the origin such that in the new coordinate zl,

0 Z1 = azl 0z I (4.31)

Proof. Write f ( z ) = az2C(z). It suffices to find a new coordinate Zl = h(z) with h(0) = 0, h'(0) = 1 such that z (1 + zqh(z)) (O/Oz) = Zl(O/Ozl). This gives a singu- lar ordinary differential equation for h. To convert this equation to a nonsingular one, write


h(z) = z~(z). The equation for h becomes

0r (1 + zqS) Oz ~r ~(0) = 1,

which has a solution. []

243

(4.32)

Proof of Theorem 4.23. (This proof is sketched in [$2].) Let X be an infinitesimal CR automorphism of M with X -- Y0 + O(1). By (4.20) and Proposition 4.5, [X, Y_,~] = - m Y _ m + O ( - m + 1). We first show that we can find an infinitesimal CR automorphism X_m of M such that [X, X_m] = - r n X _ , ~ . Suppose we have found an infinitesimal CR automorphism Xk_m of M such that [X, Xk_m] = -rnXk_,~ + X ' with X ' = O(k + 1). Then, because V is a Lie algebra, X ' C V. If X ' = 0, set X_m = X_km. Otherwise let j = w ( X ' )

and set x J m = X_k.~ - (1 / ( rn +j) )X ' . Then j > k and [X, x J ,~ ] = -mXJ_m + O(j + 1).

By induction, we can find an infinitesimal CR automorphism x_mm of M such that [X, X_~,~] = -rnX~_m + O(m + 1). By Theorem 3.3 and Proposition 4.10, Wm+l = {0}. Hence, we can

T n set X_m = X_,,~.

By Proposition 4.24 with c~ = 1 and Proposition 4.30 with a = 1, we can find new

variables (Zl, wl) such that X_m = 2Re (O/Owt), X = 2 Re(zl (O/OZl) + mwl (O/Owl)) and the equation of M has the form

vl = F,(z , ,N) (4.33)

where Fl has no pure terms. Applying X to (4.33) shows that F, is homogeneous of degree m, hence M is homogeneous.

Conversely, suppose M is homogeneous, with homogeneous part M0. Then V ~ V0 and the map To of (4.18) is an isomorphism. Let X ' = T0-1(Y0) C Wo/Wl and let X E W0 be a representative of X/. Then X = Yo + O(1). [ ]

Remark 4.34. The proof of Theorem 4.23 does not give an explicit formula for the biholomorphic map taking a homogeneous hypersurface in rigid normal form to its homogeneous part. In many cases in Sections 7-9 we find a formula for the map. [ ]

5. The form of an infinitesimal CR automorphism

Let M be a rigid hypersurface in normal form and let X be an infinitesimal CR automorphism

of M. We may assume that X is given by (4.8) and that (4.12) is the equation of M, so (4.13)

holds. Because m is rigid, the vector field X1 = 2Re 9(O)(O/Ow) is tangent to m . Hence we may assume that 9(0) = 0, by replacing X with X - Xl if necessary. In this section we derive


some properties of the functions f and g. It is convenient to write

f ( z , w ) = Z z Y f j ( w ) , 9 ( z , w ) = Z z J g j ( w ). (5.1) J J

(Note that the functions fj and gj in (5.1) are holomorphic functions of one variable and are not the same as the functions f , and g~, in (4.9), which are holomorphic polynomials in two variables. The similarity of notation should not cause any confusion.)

We begin by setting 2 ----- 0 in equation (4.13). Because F is the right side of (1.2),

~(9(Z,U)- g(0, U)) ---- ((~i01Z rrt-I "~- h i (z ) ) ~00(u). (5.2)

Setting z = 0 in (5.2) gives

By (5.2) and (5.3),

g0(u) = (5.3)

g(z ,u)=go(u)+ 2i(6io~Z "~-' +h,(z)) fo(u).

Substituting (5.3) and (5.4) in (4.13) yields

(5.4)

1 (go(u + i F ) - g o ( u - iF)) + ((5~o,Z m - ' + hi(z)) ~(u +iF) 2i

+ (6 i0 ,U" - ' + ~ ( ~ ) ) f o ( u - i F )

= ~ Fz zJf~(u + iF) + ~ E-~2J~(u - iF). (5.5)

Here we use the subscript z or 2 to denote partial differentiation with respect to z or ~ and we evaluate F, Fz, etc. at (z, 2).

Fixing u and looking at the terms in (5.5) that are homogeneous of degree m - 1 in z and gives

(5.6) (z m-' fo(u)) --pzfo(u)+ p -fo(u)

where p denotes the right side of (3.2), i.e., terms homogeneous of degree m in F . The following lemma shows that, if f0 ~ 0, equation (5.6) determines p.

L e m m a 5.7. If fo ~ O, the homogeneous part Mo of M has the form (1.4) and fo = - ~ fo. In particular, io = 1.


P r o o f . If i0 > 1, by looking at the terms in (5.6) that are homogeneous of degree r a - io

in Y and degree io - 1 in z, we see that fo --: 0. If i0 = 1 and fo ~ 0, choose u with fo(u) ~ O. Then, by (5.6), if we set co = ---fo(U)/fo(u), p satisfies

PZ __~m--I =co(p2__Zm-1). (5.8)

Hence, if m > 2, co is independent of u and f o ----- - co lo . Because al ,m-1 = 1 = am-l ,~ and a j , i = ai,j, we can solve (5.8) inductively for the coefficients ai, j of p tO see that p is given by the right side of (1.4) and com-2 = 1. [ ]

By Proposition 1.5, we may assume that, if the homogeneous part M0 of M has the form (1.4), it has the form (1.6). We make this assumption for the remainder of this section. Then,

fo = - f o if m >_ 3. (5.9)

Next we derive an equation for 9~. Write

Z b ,J (5.1o) i+j>m

L e m m a 5.11.

go(U) = io f l (u) § (m -- io) f l ( u ) + (io + 1)bio+l,m-iofo(u). (5.12)

P r o o f . Apply 0*z~ "~-*~ to (5.5) and evaluate the resulting equation at z = 0 = g. This

gives (5.12) b e c a u s e bio,m_io+l = O. []

C o r o l l a r y 5.13. If fo -- O, in particular, if io > 1,

go(U) = raRe f l (u) . (5.14)

If, in addition, io < m -- io, then f l (u) is real for u real.

P r o o f . This follows immediately from (5.3), Lemma 5.7, and (5.12). [ ]

The next lemma shows that if X is nontfivial, the right side of (1.2) simplifies.

246

Lemma 5.15.

Nancy K. Stanton

l f X ~ O ,

F(z, ~) = p(z, ~) + Izl2~~ ~)

where b = O(m - 2io - 1).

(5.16)

Proof. We use the notation of (1.2). If io = 1, there is nothing to prove. Suppose i0 > 1 and hj ~ 0 for some j . Let jo be the smallest such j . Then there are j l and a holomorphic function h such that hjo(Z ) = zJlh(z) and h(0) ~ 0. By Lemma 5.7, f0 ~ 0. Applying ozJ~o~ j~ to (5.5) and then evaluating at z = 0 = ~ gives

h(O)go(U ) = jlh(O)fl (u) + joh(O)fl (u).

Dividing by h(0) and substituting (5.14) in (5.17) gives

(5.17)

(5.18)

Now, j l > jo, so (5.18) implies that f l = 0. It follows from (5.14) that 9~ ~ 0 so, since g(0) = 0, 9o ~ 0 and, by (5.4), g ~ 0. Thus (5.5) becomes

O = ~ Fz zJ f j (u + i F ) + ~ F-~J f j (u - iF). (5.19) j>2 j>_2

Applying 0~ j~ to (5.19) and evaluating at 2 = 0 yields

0 = h}o(Z ) ~_, zJfj(u). (5.20) j>2

The sum in (5.20) is just f ( z , u). Because h'. ~ 0, we conclude that f ( z , u) ---- O, hence 30 X - O . []

Our final lemma shows that f j , j > 2, is determined by fo and fl .

Lemma 5.21. /f X ~ 0,

and for j 7s 0, 1, m

m - - 1

2 fj( ) -

m - - 1

( 5 . 2 2 )

- - (5.23)


Proof . Apply O~ i~ to (5.5) and evaluate at g = 0. By Lemmas 5.7 and 5.15, this gives

- i o ) z

+ ioz "~-i~ fZ(u) + 2 ~ bk,2z k fo(u)

-- iz2("~-i~ fo'(U). (5.24)

Now (5.22) follows by applying Oz 2m-i~ t o (5.24), using Lemma 5.7 and setting z = 0 and (5.23) follows by applying Oz rn- i~ and setting z = 0. [ ]

Corollary 5.25. If fo =- O, in particular, if io > 1, then

f ( z , w) = zf l (w) and 9(z, w) = go(w). (5.26)

Proof . This follows from (5.4), (5.22), (5.23), and Lemma 5.7. [ ]

6. Proof of Theorem 3.3

Let M be the homogeneous hypersurface given by (3.2). As in Section 5, we let p denote the fight side of (3.2). Let X be an infinitesimal CR automorphism of M. By Theorem 2.2, X = 2Re Z for some holomorphic vector field Z. Write

Z : ~ Z~, (6.1) v

where Z~ is a holomorphic vector field that is homogeneous of weight v (see Definition 4.2). Then

X = Z X , (6.2)

where X~ = 2Re Z~. Because M is given by (3.2), each X , is tangent to M and hence, by Theorem 2.2, is an infinitesimal CR automorphism of M. It follows that

x e = Z Xv and X ~ Z Xv (6.3) v = 0 modm u ~ 0 modm

are infinitesimal CR automorphisms of M. By (5.4) and Lemmas 5.7, 5.11, and 5.21, there are


holomorphic functions fo(w), f l (w), and 9o(W) such that

X ~ = 2Re z f l ( w + go(w ,

and

go : raRe f , (6.4)

X ~ = 2Re fo(w) q- - - fo'(W) + 2iz m-l -foo(W) (6.5) m - 1 Ozz "

If X is given by Theorem 3.3, then X annihilates the defining function v - p(z , -Z) of M. Hence, X is tangent to M and so, by Theorem 2.2, it is an infinitesimal CR automorphism of M. To complete the proof of Theorem 3.3, it suffices to show that X ~ and X ~ are of the form given by Theorem 3.3. We do this in Propositions 6.6 and 6.15.

Proposition 6.6. There are constants e E C and a, e E R such that

( 0 ( a + m ( R e c ) w + 2 e w 2 ) ff--~). x = 2Re \z(c + ew) +

Moreover, if m • 2i0, then c C R and e = O.

(6.7)

Proof. Because X e is tangent to M,

1 (go(u + ip) - go(u - ip)) = zpz f l (u + ip) + - ~ p ~ ( u - ip). (6.8) 2i

Apply az2i~ :(m-/~ to (6.8) and evaluate at z ---- ~ =- 0. This gives

0 = iof~(u) -- (m -- io)fl '(U). (6.9)

By (6.9), f ( = 0 if m 7~ 2/0. Also in this case, by Corollary 5.13, f l is real. Since g~ = raRe f l by (5.14), we are done if m ~ 2io. If m ---- 2io, by (6.9), f [ (u) is real for u real. Hence, h(u) = f l (u ) - f l(0) is real for u real. In this case, p = Izl ~, so (6.8) becomes

1 (go(u + ilzl m) - go(u - i lzlm)) 2i

_ m ~lzl TM ( f l (0) + h(u + ilzl ~) + f ( 0 ) + h ( u - i t z lm)) . (6.10)

By (5.14), we can write this as

1 (go(u + i[zl m) - g o ( u - ilzlm)) ---- l lzlm (gO(U + ilzl ~) + gO(U- ilzlm)). (6.11)

2i Z,


Let a = 90(0) and c = f , (0). Substitute u ---- ilz] m in (6.11)to obtain

249

9o(2i lz l "~) - a = i l z l m (go(2/Izl TM) + m R e c ) (6.12)

which becomes

t (go(t) + m R e c ) g 0 ( t ) - a = (6.13)

after the substitution t = 2ilzl m. Real solutions of (6.13) that are analytic at 0 have the form

go(t) = raet2 + ( R e c ) m t + a (6.14) 2

for some e C R. It follows from (5.14) and (6.14) that f l ( w ) = c + ew. The proposition now

follows from (6.4). [ ]

Proposition 6.15. I f X ~ ~ 0, then io = 1 and M has the form (1.4). In this case, there are constants b and d C C such that

X ~ = 2Re (b + d w + 2 id z )Oz + (2 ibz '~- ' + 2 i d z w ) . (6.16)

Furthermore, if m >_ 3 then d = 0 and -b = - ~ b .

P r o o f . The first part follows from Lemma 5.7. Suppose that i0 : 1 and M has the form

(1.4). Because X ~ is tangent to M , by (6.5) we have

f0 2i - - p)) z "~ - ' ~ (u + i p) + -2m - l f o (u - i p) = pz (u + i p) + zm f o ' (u + i m - 1

+ p~ ( u - ip) m - 1 U ' ~ f ~ i . (6.17)

We expand (6.17) in a Taylor series about u. The resulting equation for terms homogeneous of

degree 2 m - 1 in z and ~ is

- m-lpfo(u ) 2

ppz fo(U) + zmp~fo ' (u) m - 1

2 - PP~fo ' (u) - - ~ p ~ f [ ~ ( u ) .

m - 1 (6.18)


Suppose m ~ 3. By Lemma 5.7, foo' = --wf~. If f~ ~ 0, using this in (6.18) gives

p(pz + wz m-1 + wp~ + -2 "~-1) - - - 2

m - - 1 (wz,~pz + -s (6.19)

By (1.4),

1 p : - - ((wz + 2) m - (wz) m - 2 ~ ) ) . (6.20)

7T~OJ

Substituting the right side of (6.20) for p in (6.19) and examining the coefficient of z~-~ m-1 shows that (6.19) does not hold for any m > 3. Hence, if m > 3, f~ = 0, so f0 = b is constant and b -wb. If m 2, appling 3 2 : : 0z0 ~ to (6.17) and evaluating at z : 0 : ~ gives fo "(u) ---- 0, so f0 is linear. The proposition now follows from (6.5). []

7. The n o n h o m o g e n e o u s case, I: i0 > 1

The main result of this section is a complete description of the infinitesimal CR automorphisms of a rigid hypersurface M of finite type if i0 > 1. This description is contained in the following theorem.

Theorem 7.1. Let M be a rigid, analytic, real hypersurface of finite type in C 2. Suppose M is in normal form, io > 1, M is not equal to its homogeneous part, and M has a nontrivial

infinitesimal CR automorphism that is not a multiple of O/Ou. Then either M is given by the equation v = H (I z 12) for some analytic function H and

0 a 0 ) (7.2) X = 2 R e iOzoz + Ow

for some a, 0 E R or M is CR equivalent to its homogeneous part via the change of variables

z ' = z e cw, w ' : - - ~ - I (e . . . . - 1 ) (7.3) m c

or, if m = 2io, the change of variables

z' (cos cw) -2 /mz , W' 1 : : - tan cw (Tz~,.._, c

for some c E R.

As a corollary, we have the following explicit description of those M satisfying m = 2i0 that admit infinitesimal CR automorphisms in addition to multiples of O/Ou.


C o r o l l a r y 7.5. Suppose that M is a rigid hypersurface in normal form, that the homogeneous part of M is v = I zl for some m > 4, and that M is not equal to its homogeneous part. If M has an infinitesimal CR automorphism X that is not a multiple of O/Ou, then M is given by v = H( l z l 2) for some analytic H. In this case, if M has an infinitesimal CR automorphism that is not a linear combination of O/Ou and Re ( iz(O/Oz)) then M is either

1 v = - - sin -1 mclzl ~ (7.6)

m c

or

1 v = - sinh -1 a]z] m. (7.7)

a

In the first case, M is equivalent to v' = Iz'l m via (7.3) and in the second case it is via (7.4) with c = a/2.

If m r 2/o and M is obtained from its homogeneous part via (7.3), one can write down an equation for M . This is an implicit equation for v. It is easy to verify that the solution v = F ( z , ~) is an equation in rigid normal form.

By Corollary 5.25, the following result implies Theorem 7.1.

T h e o r e m 7.8. Let M be a rigid, analytic, real hypersurface of finite type in C 2. Suppose M is in normal form, M is not equal to its homogeneous part, and

X = 2Re ( z f ( w ) ~ z + 9(w) ff-~) (7.9)

is a nontrivial infinitesimal CR automorphism of M , where f and g are holomorphic functions and g(O) = O. Then either m is given by the equation v = H ( I z l 2) for some analytic function H and X is given by (7.2) with a = 0 or M is CR equivalent to its homogeneous part via the change of variables (7.3) or, tf m = 2i0, (7.4).

Proof. We use the same notation as in Section 5. We prove the theorem first in the case that m # 2i0. By Corollary 5.13, g' ~ 0. Also, by (5.3) and Corollary 5.13, because X is tangent

to M ,

2~(9(u + iF ) - 9(u - i F ) ) = (zFzg ' (u + i F ) + -fF-~9'(u - iF ) ) . (7.10)

252 Nancy K. Stanton 2i0 2m-2i0 Applying 0 z 0 3 to (7.10) and evaluating at z = 0 = 2 gives

t 262io,2m_2i0ff (u) -~- b2io,2m-2iog (tt) =

I f b2io,2m-2io ~- O, gtt(o) ~ O, and, because f ~ 0,

i ( 2 i 0 - m ) 9 , ( u ) . (7.11) rft

f =-- mE (7.12)

for some real constant c :fi O. If b2io,2m-2io ~ O, then by (7.11)

( iTFtb2i~176 u ) 9 ' (u ) = bexp \ 2 i o - m (7.13)

for some b r 0 E R. If (7.12) holds, 9 ( w ) = m e w so (7.10) becomes

1 F = - - ( z F ~ + -2F-~), (7.14)

m

which implies that F is homogeneous of degree rn. Because M is not equal to its homogeneous part, we conclude that (7.12) does not hold, so (7.13) holds. Let

Then, a r 0, a is real and by (7.13),

Substituting (7.16) in (7.10) yields

a -- imb2io,2m-2io (7.15) 2io - m

9 ( w ) = b ( e x p a w - l).

By induction

(7.16)

1 1 2~a(~iaF -- ~--iaF) ~- ~(ZFzciaF _~-~g~c-iaF). (7.17)

on the degree of the terms, equation (7.17) determines F in terms of p and b2io,2m_2i o. It remains to show that M is obtained from its homogeneous part via a change of variables of the form (7.3). Let

c -- ibuo,2m-2io (7.18) m - 2io


and let M1 be obtained from the homogeneous part v I = p(zI,U) of M via (7.3). Write the equation of MI as v = Fl(z,-s If g is given by (7.16) and X is given by (7.9) with

f = (1/ra)9/ , then X is the image of

2b (z,O 0) 2(o = - - R e + row' (7.19)

m \ Oz'

under (7.3) and hence is an infinitesimal CR automorphism of M1. It follows that F1 satisfies (7.17) and hence F = F1.

Now suppose m = 2io. Because X is tangent to M,

+ i F ) - 9 ( u - i F ) ) = ( z F z f ( u + i F ) + - 2 F - ~ f ( u - i F ) ) . (7.20)

If F(z, ~) • H(Iz l 2) for any function H , there are j r k such that bj,k r 0. Pick such a j j k and k with j + k as small as possible. Applying OzO ~ to (7.20) and evaluating at z = 0 =

yields

By (5.14), this gives

bj,k9'(u) = bi ,k( j f (u ) + k-f (u)). (7.21)

= - f . (7.25)

(7.22)

which is impossible if f ~ 0. Because X is nontrivial, (5.14) implies that f ~ 0 so we conclude that F(z , 2) = H(Iz l 2) for some analytic function H. Hence, (7.20) becomes

1 2~(9(u + i l l ) - g(u - i l l ) ) = Izl2(f(u + i l l ) + -f(u - i H) )H ' (7.23)

where H and H ' are evaluated at [zl 2. Suppose there is no analytic function G such that H([z[ 2) = G(Izlm). Let k be the smallest positive integer such that 2k ~ 0 mod rn and H (k)(0) • 0. Applying k k 0 z 0-~ to (7.23) and evaluating at z = 0 = 2 gives

H(k)(O)9'(u) = kH(k)(O)(f(u) + f (u ) ) . (7.24)

By (5.14), this implies that 9 / -- 0, so 9 -= 0, and


Using (7.25) in (7.23) gives

o = I z l 2 n ' ( f ( u + i l l ) - f ( u - i n ) ) . (7.26)

Because M is in normal form, H (m) (0) = 0. Thus, applying 0zm0 m to (7.26) and evaluating at z = 0 = ~ yields

f ' (u) = 0 (7.27)

so, by (7.25),

X = 2Re iOz~-~ (7.28)

for some 0 E R.

Finally, suppose H(Izl z) = G(Izl "~) for some analytic function G. Because M is in normal form, G(z) = z + O(z3). In terms of G, (7.20) is

~ ( 9 ( u § 2 [ z l m G ' ( I z l m ) ( f ( u + i G ) w f ( u - i G ) ) (7.29)

where G and G' are evaluated at Izl m. Let t = G([zlm). Then (7.29) becomes

~ ( g ( u + it) - g(u - it)) = 2 r162 + it) + -f(u - it)). (7.30)

where we have written Izl m = r The analytic function r satisfies r = t -t- O(t3). Let

hi (u, t) = Im 9(u + it), h2(u, t) = raRe f ( u + it) (7.31)

and

By (7.30),

h3(t) = r162 = t + O(t3). (7.32)

hl(u, t ) ---- h2(u,t)h3(t). (7.33)

The functions h~ and h2 are harmonic, so applying the Laplacian in the variables u, t to (7.33) gives

0 = h2(u,t)h~l(t) + 20th2(u,t)h;(t). (7.34)

If h2 -= 0, then (7.25) holds and, by (7.33), g -- 0 so (7.26)-(7.28) hold. If h2 ~ 0 but Oth2(u, t) =_ O, then by (7.34), h~' - 0, so, by (7.32), h3(t) = t. Because M is not equal to its


homogeneous part, we conclude that if h2 ~ 0, then Oth2 ~ O. Thus, by (7.34),

255

h'~'(t) O, h2(~, t) - - 2

h'~ (t) h2 (u, t) = -2Or(log hz(u,t)) . (7.35)

Hence, the right side of (7.35) is a function of t alone, so

h2(u, t) = r (t)r (7.36)

for some functions qSi and r Because h2 is harmonic, (7.36) implies that

qS' l' = cr (7.37)

for some c E R. By (7.32), h~(0) = 1 and h~'(0) = 0, so, by substituting (7.36) in (7.34) and evaluating at t = 0, we see that r (0) = 0. Thus, by (7.37), if we set b = x / ~ ,

acosbt , c < 0 (7.38) r acoshbt , c > O

for some a E R. Substituting (7.38) in (7.35) and integrating gives

f (cos~t) -2, c < 0, h~(t) (7.39)

(cosh bt) -2, c > 0.

Hence

1 h3(t) = g tanbt, c < 0, (7.40)

1 g tanh bt, c > O.

Thus, by (7.32) and (7.40), setting s ---- Izl TM, we have

1 tanbG(s), c < 0, s a ' ( s ) =

1 tanhbG(s), c > O. (7.41)

Now G(0) = 0, so the solution of (7.41) is

1 G(s) = g sin -1 bs, c < 0,

1 sinh - l b s , c>O. (7.42)


In the first case, M has the form (7.6) and is equivalent to its homogeneous part via (7.3) with c replaced by b/m in (7.3). In the second case, M has the form (7.7) and is equivalent to its homogeneous part via (7.4) with c replaced by b/2 in (7.4). [ ]

8. The n o n h o m o g e n e o u s case, II: i0 = 1 and m > 2

Suppose i0 = 1 and M is not strictly pseudoconvex, i.e., m > 2. In this section, we describe the space V of infinitesimal CR automorphisms of M. We treat the case that the homogeneous part M0 of M is not equivalent to a tube in Theorem 8.1. This theorem follows immediately from the results of Sections 5 and 7. Our results in the case that M0 is equivalent to a tube are given in Theorem 8.3. The key ingredient in the proof is Theorem 4.23.

T h e o r e m 8.1. Let M be a real analytic, rigid hypersurface in C 2 with io = 1 and m > 2.

Suppose M is in normal form and that its homogeneous part Mo is not equivalent to a tube.

I f M ~ Mo and M has a nontrivial infinitesimal CR automorphism X that is independent of O/Ou, then M is equivalent to Mo via the change of variables

1 z ' = ze ~ , w ' = (e . . . . -- 1) (8.2)

g r t c

for some e E R.

Proof . By Lemma 5.7 and Corollary 5.25, after subtracting a multiple of O/Ou from X if necessary, we may assume that X has the form (7.9) with g(0) = 0. The result now follows from Theorem 7.8. [ ]

For the remainder of the section, we assume that M is in normal form and that Mo is equivalent to a tube. By Proposition 1.5, we may assume that M0 is given by (1.6). Because M is rigid, Re (O/Ow) E V , so dim V > 1. By Corollary 4.15 and Theorem 3.3, dim V _< 3.

T h e o r e m 8.3. Let M be a real analytic, rigid hypersurface in C 2 with io = 1 and m > 2.

Suppose M is in normal form and that Mo is given by (1.6), and hence is equivalent to a tube.

I f M has a nontrivial infinitesimal automorphism X that vanishes at O, then M is homogeneous

and dim V = 3. I f dim V = 3 then M is homogeneous.

Proof . Suppose X is a nontrivial infinitesimal automorphism of M with X(0) = 0. By Proposition 4.10 and Theorem 3.3, the lowest weight part X0 of X is given by

X o = 2 R e CZ ~z + mCw (8.4)


for some c r 0 C R. Thus, X ' = (1 / c )X = Y0 + O(1) where Y0 is given by (4.19). Hence, by Theorem 4.23, M is homogeneous. It follows from Theorem 3.3 that dim V = 3.

Suppose dim V = 3. By Proposition 4.10 and Theorem 3.3, the maps Tj of (4.18) are isomorphisms. Let X = T0-1(Y0). By Theorem 3.3 and Proposition 4.10, W1 = {0}, so X E W0. The last statement in the theorem now follows from the first because X is a nontrivial infinitesimal CR automorphism of M that vanishes at 0. [ ]

9. The strongly pseudoconvex case

Let M be a rigid, real analytic, strongly pseudoconvex hypersurface in rigid normal form. In this section we describe the space V of infinitesimal CR automorphisms of M. The results

are summarized in Theorem 9.45.

Some of the results in this section follow from the work of Cartan [C] and Belo~apka [B]. Cartan observed that if dim V > 3, M is locally equivalent to the hyperquadric v = Izl 2. Belo~apka showed that the stability group of a point on m has dimension at most 1 unless M is locally equivalent to the hyperquadric. Our results give more precise information about the form of the infinitesimal automorphisms of a hypersurface in rigid normal form.

Our first two propositions are analogues of Theorem 8.3. In the proofs, we exploit the Lie algebra structure of V. The homogeneous part M0 of M is the hyperquadric v = lzl 2. By Propositions 4.5 and 4.10, we will be able to obtain some information about the Lie algebra structure of V from the structure equations for V0, the space of infinitesimal CR automorphisms

of Mo. By Theorem 3.3, we may take {X_2, X_I , X'_l, Xo, X~, X1, X(, X2} as a basis of Vo where Xi = 2Re Zi, X~ = 2Re Z~, and

0 Z_ 2 B

Ow'

0 0 Z-1 - Oz + 2iz o~-w'

.0 ,9 z ' , = *-07z + 2z0--7'

0 0 Zo = z-~z + 2W-~w,

0 a o = i z - -

Oz'

2 0 0 Z 1 = (W .Off 2 i z )Ozz + 2izw-ffw'

2 0 0 Z~ = (iw + 2z ) Ozz + 2ZW Ow,

o w2 0 Z2 = z w ~ + Ow" (9.1)


The vector fields Xi and X~ are homogeneous of weight i. The structure equations are

[X_2, X_I] = 0 = [X_2, Xt 11 = [X_2,Xo] , [X_2, Xo] = 2X_2,

[X_2, X1] = X_l , [X_2, X~] = X t 1, [X_2, X2] = Xo,

[X_, ,X'_ , ] = 4X_2, [X_, ,X~] = X'_,, [X_, ,Xo] = X - l ,

[X_l, X1] = 6Xo, [X_,, X~] = 2Xo, [X-l, X2] = X1,

[X'_I,Xo] = - X _ , , [X'_l,X0] = X'_,, [X'_l,Xi] = -2Xo,

[X'_,,X~] = 6Xo, [Xt_I,X2] = X{, [Xo, Xo] ~-- 0,

[X~,X,] = - x ~ , [x~,x~] = x , , [Xo, X2] = o ,

[x0,xl] = x1, [Xo, X~] = x~, [Xo, X2] = 2x2,

[x, ,x~] = 4x2, [ x ~ , x 2 ] = o = [ x ~ , x 2 ] . (9.2)

Propos i t ion 9.3. Suppose M has a nontrivial infinitesimal automorphism X whose lowest weight part is Xo, aXl + bX~ for some a, b E R, or X2. Then M is equivalent to the hyperquadric and dim V = 8.

Proof . Suppose X = X2 + 0(3) . By Proposition 4.5 and (9.2), [X_2, X] = X 0 n t- O(1). Suppose X = aX1 + bX~ + 0 (2 ) , By Proposition 4.5 and (9.2),

[X_2, X] [ [ X - 2 , X ] , X ]

[[IX_2, X], X], [X_2, X]]

[[[x_2,x],x],xl

= aX_l + bXt_l + 0 ( 0 ) ,

= 6(a 2 +bz )X~ + O ( 1 ) ,

= 6(a 2 + bZ)(bX_l - aX'_l) + 0(0),

= 6(a 2 + bZ)(bX1 - aXe) + 0 ( 2 ) . (9.4)

By Corollary 2.4, because X_ 2 E V, there are vector fields W/ E V with lowest weight parts Xi, i = - 1, 1, and W" with lowest weight parts X~, i = - 1,0, 1. Also by Proposition 4.5 and (9.2), [W-l , W~] = 2Xo + O(1).

Thus, under the hypothesis of the proposition, there is a vector field Y E V with lowest weight part X0. By (4.19) and (9.1), Y = Y0 + O (1). The proposition now follows from Theorem 4.23 and Theorem 3.3. [ ]

Proposition 9.5. Suppose M has an infinitesimal CR automorphism X with lowest weight part Xo + axe, for some a ~ 0 C R. Then M is equivalent to the hyperquadric.

First we prove a lemma.


L e m m a 9.6. Suppose M has an infinitesimal CR automorphism

X -- 2Re a Z ~ z z + 2 (Rea )w + O(1) (9.7)

with Im a r 0 such that X and X -2 generate a subalgebra V I, dim V / > 3, whose only elements of nonnegative weight are multiples of X . Then M has an infinitesimal automorphism V-2 --- X-2 -1- O ( - 1 ) such that [V-2, X] = 2(Re oz)Y_2.

Proof. Write oz = ql -[- ia2 with aj C R. By (9.2) and Proposition 4.5, there is Y r 0 E V' with w ( Y ) = - 1 such that

[X_z,X] = 24,X_2 + Y. (9.8)

Because Y = bX_l + cXt_l + O(0) for some b, c E R with b 2 + c 2 r 0, by (9.2)

[Y, X] = (alb - azc)X_l + (azb + alc)Xt_l + O(0). (9.9)

Hence, dim V' = 4 and there are ~'-1 and Y-~I E V' with Y-1 = X_t + O(0) and Y-~I =

X'_ 1 + O(0) such that Y = b~'-i + cY" 1. By (9.2),

[~'-1, X] = a1~'-1 + a2Y' , +/3)(- (9.10)

and

for some fl, 7 E R. Set

and

x] = + + -yx

- a 2 - y ) Y _ , = ~ ' _ , + \ a 2 ~ a 2 ) X

(9.11)

(9.12)

Y-tl = y-tl '~ ~ 02 ~_ (~2 / X, (9.13)

so []I-1, X] --- alY-1 +a2Yt_l and [Y-~ 1, X] = -azY-1 + alY_~ 1 �9 By (9.2) and Proposition 4.5, [Y-l, Y-~I] = 4X_2 + O(1). Let

Y-2 = 1[y-1, y_,d. (9.14) 4


Then, by the Jacobi identity applied to Y-t , Y-~t, and X ,

[]7-2, X] = 2a,Y_2. [] (9.15)

P r o o f of Propos i t ion 9.5. By Proposition 9.3, we only need to prove the proposition in the case that X_2 and X generate a subalgebra V' of V whose only elements of nonnegative weight are multiples of X . First we consider the case that dim V' = 2. Then, by (9.2) and

Proposition 4.5,

IX-2, X] = 2X_2 + o~X (9.16)

for some c~ C R. Write X = 2Re Z with Z holomorphic. By (9.16), Z has the form

,~,,,0 0 z = c(z) + 9(z, w) ffw (9.17)

where c is holomorphic and c(0) = 0. By Corollary 5.25, X has the form (7.9). Applying Theorem 7.8, we see that M is equivalent to its homogeneous part.

It remains to consider the case that dim V' > 3. The hypotheses of Lemma 9.6 are satisfied with c~ = 1 + ia. Hence, we can find Y C V' with lowest weight part X - 2 such that [Y, X] = 2Y. By Proposition 4.24, we can find coordinates (z' , w') such that Y = 2Re (O/Ow'), m has an equation of the form

v' = G(z',-~') (9.18)

where G has no pure terms and

X = 2Re ((1 + ia)z' + f(z ')) Oz--- 7 + 2w' (9.19)

where f = 0 ( 2 ) is holomorphic. By Proposition 4.30, we can find new coordinates z" = h(z'), w" = w' such that X = 2Re ((1 + ia)z"(O/Oz") + 2w"(O/Ow")). In these coordinates, M is given by an equation of the form

v " = H(z",~") (9.20)

where H has no pure terms. Applying X to H shows that H is a function of Iz[ 2 that is homogeneous of degree 2. [ ]

By Propositions 9.3 and 9.5, if dim V < 8, then dim V < 4. For our analysis of the case that dim V < 4, we need information about the case that the equation of m is invariant under rotations in z. If M is given by an equation of the form v = F(]zl2), then 2Re (O/Ow), 2Re iz(O/Oz) C V so dim V _> 2. The next two propositions analyze this case.


Proposit ion 9.21. Suppose that M is given by the equation

v = F(]z] 2) (9.22)

with F in normal form. Then either dim V = 2 or F is given by one of the functions

Iz12' 271 sin- 12clzl2 ' 2~1 sinh- l 2clz[2 (9.23)

for some c E R. If F is given by one of the functions in (9.23), then M is homogeneous.

R e m a r k 9.24. If M is given by one of the last two functions, then it is equivalent to its homogeneous part via a change of variables of the form (7.3) or (7.4). [ ]

Proof. Write X = 2Re Z with Z holomorphic, and write

Z - - Z Z~ (9.25) / /

where Z~ is a holomorphic vector field homogeneous of weight u. Set

Z e = Z Z~, Z ~ Z Z~. (9.26) v e v e n u o d d

Then, because the equation of M has the special form (9.22), X e = 2Re Z * and X ~ = 2Re Z ~ are tangent to M and either X * or X ~ is nontrivial. Suppose X * ~ 0. By Corollary 5.25,

o Z ~ = z f ( w ) + 9(w) O-w (9.27)

for some holomorphic functions f and 9. It follows from Theorem 7.8 that either dim V = 2 or F is given by one of the functions in (9.23).

Suppose X ~ ~ O. By (5.4) and Lemma 5.21, because the equation of M is in normal form,

Z~ = ( f (w) + 2iz2-f ' (w) ) O + 2iz-f (w) ff-- w (9.28)

for some holomorphic function f with f ~ 0. Because X ~ is tangent to M,

z-f(u + iF) + ~ f ( u - iF) = ( ~ f ( u + iF) + 2iz2-2 f ' ( u + iF)

+ z-f(u - iF) - 2iz-22f'(u - iF) ) F' (9.29)


with F and F evaluated at lzl 2. Equating the terms in (9.29) with one more 2 than z yields

f ( u - i F ) = ( f ( u + i F ) - 2ilzl2f'(u - i F ) ) F ' . (9.30)

Because M is in normal form, F ' (0 ) = 1, F" (0 ) = 0. Let t = ]z[ 2. Apply 0 2 / 0 t 2 to (9.30)

and evaluate at t = 0. This gives

Hence

2 f " ( u ) = f ( u ) F ' " ( O ) . (9.31)

[]

(9.37)

In the next proposition we complete our analysis of the case that there is a nontrivial vector field X with w ( X ) = O.

substituting (9.32) in (9.33) yields

Co = (Co + 2 i c l F - 2 ic l t ) F ' .

The solution of (9.37) is F ( t ) = t, so M is the hyperquadric.

c o c o s c u + c l s i n c u , c = ~ , F ' " (0 ) < 0

f(u) = Co + ClU, F " ( 0 ) = 0 (9.32)

cocoshcu + cl sinhcu, c = V / ~ , F " ( 0 ) > 0.

We let u = i F in (9.30). The equation becomes

f (O) = ( f ( 2 i F ) - 2 i l z l 2 f ( O ) ) F ' . (9.33)

Suppose F" ' (0 ) < 0. We substitute (9.32) in (9.33) to obtain

Co = F 'co cosh 2 cF + c l i (sinh 2 cF - 2ct) F ' (9.34)

where again we let t = ]zl 2. The unique solution F of (9.34) with F (0 ) = 0 is

F ( t ) = 1 sinh-' 2ct, (9.35) 2c

and M is equivalent to the hyperquadric via the change of variables (7.4). Similarly, if F ' " > 0, the solution of (9.33) is

F ( t ) = 1 sin- l 2ct, (9.36) 2c

and M is equivalent to the hyperquadric via the change of variables (7.3). Finally, if F ' " (0) = 0,


Proposi t ion 9.38. Suppose dim V _< 4 and that there is a nontrivial X E V with lowest weight part X~. Then the equation of M has the form

v---- F( lz l ~) (9.39)

for some real analytic function F of one variable and dim V = 2.

Proof. Let V' be the subalgebra of V generated by X_2 and X. Suppose dim V' = 2. We take {X_2, X} as a basis of V. By (9.2),

[X_2, X] = a X (9.40)

for some a E R. Write

X = 2 R e f ( z , w ) - ~ z OwJ" (9.41)

By (9.40), O f l O w = a f , so f ( z , w) = c(z)e aw. Because the lowest weight part of X is X~, c(0) = 0. By Corollary 5.25, X has the form (7.9). By Theorem 7.8, because dim V < 8, the equation of M is given by (9.39).

Now suppose dim V' > 2. By Propositions 9.3 and 9.5, the hypotheses of Lemma 9.6 are satisfied with ce = / . Hence, we can find Y E V' with lowest weight part X_2 and [Y, X] = 0. By Propositions 4.24 and 4.30, we can make a change of coordinates so that in the new coordinates (z', w'), X = 2Re iz(O/Oz) and M has an equation of the form

v' = /W(z', ~') (9.42)

where F has no pure terms. Applying X to (9.42), we see that the equation has the form

v ' = F'(Iz'12). (9.43)

By Remark 1.17, (9.43) can be put in normal form

v " = F"(Iz"[ 2) (9.44)

by a change of coordinates of the form z' = z"e i~ w' = w".

If dim V > 2, it follows from Proposition 9.21 and the fact that M has an equation in normal form given by (9.44), that M is homogeneous and hence dim V = 8. Because dim V < 4, we conclude that dim V = 2 = dim V', so the equation of M is of the form (9.39). [ ]


We summarize the results of Propositions 9.3, 9.5, 9.21, and 9.38 in the following theorem.

Theorem 9.45. Let M be a strictly pseudoconvex hypersurface in rigid normal form. Let V be the space of infinitesimal CR automorphisms of M. Then, either dim V : 8 and M is equivalent to the hyperquadric or dim V _< 3. If dim V ~ 3 and there is a vector field X E V with X : X ~ + O ( 1 ) , then dim V : 2 and M is given by an equation of the form v : F(Izl2) If dim V = 3, there is no nontrivial X E V with w ( X ) >_ O.

Proof . By Propositions 9.3 and 9.5, if there is a nontrivial X E V with lowest weight part Xo, aX~ + Xo, aX1 + bX~, or X2, then M is equivalent to the hyperquadric. Hence, if M is not equivalent to the hyperquadric, dimV _< 4. Also, if dim V _< 4 and there is a nontrivial X E V with w ( X ) > O, then the lowest weight part of X is X~. In this case, by Proposition 9.38, dim V : 2 and M is given by an equation of the form v ----- F(Izl2). Thus, if M is not equivalent to the hyperquadric, dim V _< 3. The last statement in the theorem is an

immediate corollary of Propositions 9.3, 9.5, and 9.38. [ ]

10. Infinitesimal approximate dilations and rotations

The results of Sections 4 and 7-9 allow us to give geometric characterizations of homogeneous and rotationally invariant hypersurfaces. Before we state the main results of this section, we introduce some definitions.

The one-parameter group of holomorphic transformations of C 2 generated by the vector field

is the group of rotations

X = 2ReiOz 0 (10.1)

Ct(z, w) = (eit~ z, w). (10.2)

The group generated by the vector field

Y ---- 2Re c z ~ z + mew , (10.3)

where c r 0 E R, is the group of (nonisotropic) dilations

Ct(z, W) = (eCtz, emctw). (10.4)

Thus, we call X an infinitesimal rotation and Y an infinitesimal dilation.

The geometric characterizations use the following approximate versions of infinitesimal dilations and rotations.


Definit ion 10.5. Let M be a real analytic, rigid hypersurface through the origin in C 2 and let W be an infinitesimal CR automorphism of M. We call W an infinitesimal approximate rotation if

W = 2ReiOzOw - + O(1) (10.6) O Z

for some 0 ~ 0 E R. We call W an infinitesimal approximate dilation if

W ~ - 2 R e cz-~z + m e w + 0 ( 1 ) (10.7)

for some c 7 ~ 0 C R.

Thus, if W is an infinitesimal approximate rotation, W has a fixed point at the origin with linearization X + 2 R e aw(O/Oz) for some a C C. Similarly, if W is an infinitesimal approximate dilation, W has a fixed point at the origin with linearization Y + 2Re aw(O/Oz) .

Hypersurfaces of the form v = F([z[ 2) are invariant under the one-parameter group of

rotations (10.2).

Definition 10.8. A rigid hypersurface M is rotationally invariant about the origin if it is locally equivalent to a hypersurface v ~ z F ( z'~ ~ ) that is invariant under the one-parameter group of rotations (10.2).

Our first theorem characterizes rotationally invariant hypersurfaces.

T h e o r e m 10.9. Let M be a real analytic, rigid hypersurface in C 2. Suppose that M is in rigid normal form. Then M is rotationally invariant if and only if M has an infinitesimal approximate rotation. Furthermore, if m is rotationally invariant and is not equivalent to the hyperquadric Imw = ]z[ 2, then M is given by an equation of the form Imw = F(lz[2).

Proof . Suppose M is rotationally invariant. Let q5 : M ~ M ~ be a biholomorphic equivalence preserving the origin, where M ~, given by v ~ = F ( z ~, ~7), is invariant under (10.2). Then the vector field X in (10.1) is an infinitesimal CR automorphism of M ~, and qS~-lX is an infinitesimal approximate rotation of M.

If M has an infinitesimal approximate rotation, by Theorem 3.3 and Proposition 4.10, the

homogeneous part of M is

v = ( 1 0 . 1 0 )

The theorem now follows from Corollary 7.5 and Theorem 9.45. [ ]

266

C o r o l l a r y 10.11.

equation of the form

Nancy K. Stanton

Let M be a real analytic, rigid hypersurface in C 2 given by a rigid

v = p ( z , - f ) + O ( m + 1) (10.12)

where p is a homogeneous polynomial of degree rn having no pure terms. Then M is rotationally invariant if and only if M has an infinitesimal approximate rotation.

P r o o f . By Remark 1.17, M can be transformed into rigid normal form by a map of the

form (z, w) = 69(z', w ' ) = ( ( a z ' + f(z ' ) )ei~ + w ' + g(z ' ) ) with f , g = 0 ( 2 ) . Because p

has no pure terms, 9 = 0 ( 3 ) . Hence �9 and its inverse take an infinitesimal approximate rotation

to an infinitesimal approximate rotation. The corollary now follows from Theorem 10.9. [ ]

Hypersurfaces of the form

I m w = p ( z , ~ ) , (10.13)

where p is a homogeneous polynomial, are invariant under the one-parameter group of dilations

(10.4).

The following theorem characterizes homogeneous hypersurfaces. This answers Rothschild's question in C 2.

Theorem 10.14. equation of the form

Let M be a real analytic, rigid hypersurface in C 2 given by a rigid

v = p(z,-2) + O ( m -q- 1) (10.15)

where p is a homogeneous polynomial of degree m having no pure terms. Then M is homogeneous if and only if M has an infinitesimal approximate dilation.

Proofi As in the proof of Corollary 10.11, we see that a transformation taking M into

rigid normal form takes infinitesimal approximate dilations to infinitesimal approximate dilations.

The result now follows from Theorem 4.23. [ ]

[BJT]

[BRTI

[B]

References

Baouendi, M. S., Jacobowitz, H., and Treves, E On the analyticity of CR mappings. Annals Math. 122, 365-400 (1985). Baouendi, M. S., Rothschild, L. P., and Treves, E CR structures with group action and extendability of CR functions. Inventiones Math. 82, 359-396 (1985). Belo~apka, V. K. On the dimension of the group of automorphisms of an analytic hypersurface. Izv. Akad. Nauk SSSR Ser. Mat. 43, 243-266 (1979); Math. USSR Izvestija 14, 223-245 (1980).


[C] Cartan, E. Sur la gromrtrie pseudo-conforme des hypersurfaces de l'espace de deux variables complexes, Part I. Annali di Mat. 11, 17-90 (1932); reprinted in his Oeuvres Completes, Vol. II, 1231-1304. Paris: Gauthiers-Villars 1953 (reprint, Paris: Editions du CNRS 1984).

[D] D'Angelo, J. E Defining equations for real analytic real hypersurfaces in G 'n. Trans. AMS 295, 71-84 (1986). [KN] Kobayashi, S., and Nomizu, K. Foundations of Differential Geometry, Vol. If. New York: Wiley Interscience

1969. [K] Kruzhilin, N. G. Description of the local automorphism groups of real hypersurfaces. In: Proceedings of the

International Congress of Mathematicians, Berkeley (1986), Vol. I, 749-758. [MZ] Montgomery, D., and Zippin, L. Topological Transformation Groups. New York: Wiley Interscience 1955. [S1] Stanton, N. K. A normal form for rigid hypersurfaces in C 2. Amer. J. Math. To appear. [$2] Stanton, N. K. Rigid hypersurfaces in C 2. In: Proc. Symp. Pure Math., Vol. 52 Part 3, 347-354. Providence:

Amer. Math. Soc. 1991. [T] Tanaka, N. On the pseudo-conformal geometry of hypersurfaces of the space of n complex variables. J. Math.

Soc. Japan 14, 397-429 (1962).

Received March 30, 1990

Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556

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Infinitesimal CR automorphisms of rigid hypersurfaces in C2