SCIENCE CHINA Mathematics

. ARTICLES .

March 2010 Vol. 53 No. 3: 687–700 doi: 10.1007/s11425-010-0047-1

Equivalence problem for Bishop surfaces Dedicated to Professor Yang Lo on the Occasion of his 70th Birthday

HUANG XiaoJun1,∗ & YIN WanKe2 1Department 2School

of Mathematics, Rutgers University, New Brunswick, NJ 08902, USA; of Mathematics and Statistics, Wuhan University, Wuhan 430072, China Email: [email protected], [email protected] Received September 14, 2009; accepted December 23, 2009

Abstract The paper has two parts. We first briefly survey recent studies on the equivalence problem for real submanifolds in a complex space under the action of biholomorphic transformations. We will mainly focus on some of the recent studies of Bishop surfaces, which, in particular, includes the work of the authors. In the second part of the paper, we apply the general theory developed by the authors to explicitly classify an algebraic family of Bishop surfaces with a vanishing Bishop invariant. More precisely, we let M be a real submanifold of C2 defined by an equation of the form w = zz + 2Re(z s + az s+1 ) with s  3 and a a complex parameter. We will prove in the second part of the paper that for s  4 two such surfaces are holomorphically equivalent if and only if the parameter differs by a certain rotation. When s = 3, we show that surfaces of this type with two different real parameters are not holomorphically equivalent. Keywords equivalence problem, Bishop surface, Chern-Moser theory, elliptic and hyperbolic complex tangents, normal form and modular space MSC(2000):

20M31, 33Q21

Citation: Huang X J, Yin W K. Equivalence problem for Bishop surfaces. Sci China Math, 2010, 53(3): 687–700, doi: 10.1007/s11425-010-0047-1

1

Introduction

An important problem in several complex variables and complex geometry is to classify complex manifolds, CR manifolds, etc., under the action of biholomorphic maps. A basic approach to study the equivalence problem is to attach a complete set of invariants to the objects. For compact complex manifolds, the invariants are often integral invariants (e.g., Chern class and others). However, for non-compact complex manifolds, integral invariants are often trivial or hard to compute. One then has to look for other invariants. When manifolds have smooth boundaries and its boundaries have certain convexity, the equivalence problem can be reduced to the boundary CR equivalence problem. To illustrate such an idea, we first state the following well-known theorem in several complex variables (For the definition of the basic concepts involved, we refer the reader to the book [2]): Theorem 1.1 [12, 19]. Let D1 and D2 be two bounded strongly pseudoconvex domains in Cn with C ∞ boundaries. Assume n  2. Then D1 and D2 are biholomorphically equivalent if and only if there is a smooth CR equivalence map from ∂D1 to ∂D2 . ∗ Corresponding

author

c Science China Press and Springer-Verlag Berlin Heidelberg 2010 

math.scichina.com

www.springerlink.com

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This theorem gives a reduction of the domain classification problem to the global boundary CR classification problem. When the domains have real analytic boundaries, one can further apply the holomorphic continuation property for holomorphic maps along curves (see [37]) to reduce the situation to the following local equivalence problem for real submanifolds in a complex manifold, which we state in its most general version: Problem 1.1. Let M1 and M2 be two real submanifolds in Cn . When does there exist a biholomorphic map F defined near M1 such that F (M1 ) = M2 ? We notice that the problem is an existence problem: Given two real submanifolds in Cn , the problem asks when one can find a biholomorphic transformation sending one to the other. One may formulate it as a functional equation with solutions satisfying the d-bar equation. However, people rarely use such a method to approach such a problem. In the following, we will consider two cases: the hypersurface case and the case of Bishop surfaces (generically embedded real surfaces in C2 ). For the study of the equivalence problem in the other situations, we refer the reader to [2] and a recent paper of the authors [25], as well as the references therein.

2

Strongly pseudoconvex hypersurfaces

In this section, we briefly describe the Chern-Moser theory [9] for strongly pseudoconvex hypersurfaces. There are two basic ways to construct invariants in this setting. One is the normal form theory and the other is the geometric Cartan theory. 2.1

Normal form theory

The normal form theory is classical and has been used in many branches of mathematics for the classification problem. To illustrate the idea, we first look at an example from linear algebra. Let A, B be two n × n real symmetric matrices. Say A and B are equivalent if there is an orthogonal matrix P such that P AP t = B. We then ask how to classify real symmetric matrices under such an equivalence relation. If we directly approach the problem, we then have to solve the system of non-linear equations P AP t = B for P , which is un-doable unless A and B are extremely simple. The way to approach the problem is to transform the matrices by orthogonal transformations into diagonal matrices, called the normal form under this relation. Then the diagonal elements, called eigenvalues, uniquely determine the equivalence class of the matrices. Namely, the set of eigenvalues form a complete set of the invariants in this situation. In this process, to obtain the normal form and the map sending the original matrix to the normal form, one needs to solve finitely many linear equations of the form: (λI − A)x = 0. In this example, the normalization procedure ends after finite many steps. In many other problems, the normal form can not be completed in finitely many times. Then the normal form is usually presented by a special type of (formal) power series. Various convergence problems thus jump in. Since the invariants are now embedded into the power series, or the derivatives of a certain special representation function, the invariants are called differential invariants. The strongly pseudoconvex case falls into this category. We next briefly present the Chern-Moser normal form for a strongly pseudoconvex hypersurface in n+1 : C Write (z, w) = (z1 , . . . , zn , u + iv) ∈ Cn × C1 for its coordinates of Cn+1 . Let M be a real analytic strongly pseudoconvex hypersurface near 0 ∈ Cn+1 , defined by    n n  |zi zj zk | + |zi u| + |u|2 . v = |z|2 + O i,j,k=1

i=1

Set wt(zi ) = wt(z i ) = 1 for 1  i  n and wt(w) = wt(w) = 2. When n > 1, there is a unique transformation from the group    nor n Bih0,cm (C , 0) := (z  , w ) = f1 (z, w), . . . , fn (z, w), g(z, w) : f = z + Owt (2),

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 2   ∂ g ∂f (0) = 0, Re (0) = 0 , ∂w ∂w2

such that in the new coordinates, called the Chern-Moser normal coordinates (z  , w = u + iv  ) for M , M is defined by an equation of the following special form:  Fkl (z  , z  , u ). v  = |z  |2 + k,l2

Here Fkl (tz, sz, u) = tk sl Fkl (z, z, u) for any t, s > 0 and Fkl (z, z, u) further satisfies the following: trF22 = (tr)2 F23 = (tr)3 F33 = 0. n 2 Here tr is a second order differential operator defined by tr = i=1 ∂z∂i ∂zi . 2 When n = 1, there is a unique transformation from the group Bihnor 0,cm (C , 0) such that in the new     coordinates called the Chern-Moser normal coordinates (z , w = u + v ) for M , M is defined by an equation of the following special form:  k akl (u )z  z l . v  = |z  |2 + k,l2,k+l6

The normal form stated above is still subject to the action of n+1 Bih(Cn+1 , 0)/Bihnor , 0), 0,cm (C

which is precisely Aut0 (Hn+1 ) = SU (n + 1, 1). Due to the arbitrariness of the coefficients in the expansion of the normal form, we see the space of invariants can be roughly described as an infinite dimension space . a finite dimension group As an immediate application, we see that there are infinitely many invariants. We mention that an important part of the above stated Chern-Moser theory is that the normal form is always presented by a convergent power series. 2.2

Geometric Cartan-Chern-Moser theory

There is a different way to construct a complete set of invariants for germs of smooth strongly pseudoconvex hypersurfaces. Here the invariants are the Cartan curvature functions and their covariant derivatives defined on a certain G-structure bundle over M . We skip the discussion and refer the reader to the original paper of Chern-Moser [9] or the lecture notes of the first author [22].

3

Geometry associated with elliptic Bishop surfaces

A strongly pseudoconvex hypersurface in C2 near 0 is defined, after a change of coordinates, by an equation of the following form: Im(w) = |z|2 + o(2). We now consider a submanifold M defined near 0 ∈ C2 by an equation of the form w = |z|2 + o(2). Then the following can be easily verified: (1) It is a real surface of codimension 2 in C2 . (2) It has a CR (1,0) singularity at 0 in the sense that Tq M has dimension 0 for q = 0 and 1 for q = 0.

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Consider more generally a second order harmonic perturbation of the above surfaces. After a change of coordinates, we get a surface defined by an equation of the following form: w = |z|2 + λ(z 2 + z 2 ) + o(|z|2 ),

0  λ  ∞.

Such a surface is now called a Bishop surface, which was first systematically studied by Bishop in 1965 [7]. λ is the lowest order biholomorphic invariant, called the Bishop invariant of the surface. When λ = ∞, M is understood as w = z 2 + z 2 + o(|z|2 ). We call p an elliptic, hyperbolic or parabolic point of M , according to whether λ ∈ [0, 1/2), λ ∈ (1/2, ∞] or λ = 1/2, respectively. The geometric and holomorphic structure depends strongly on the nature of λ. The reason that people care about such surfaces is at least due to the following facts: • They are the generic and important model of manifolds with (CR) structure singularity at the point under study. • They have various fundamental connections with the complex Plateau problem by the work of many people (see [5, 6, 11, 22], etc.). • They are closely connected to the study of many problems in symplectic geometry, after the work of Gromov [17], Eliashberg [11], Hofer [18] and other mathematicians. • They are closely related to problems in celestial mechanics, small divisor problems in classical dynamics, due to the work of Moser-Webster [32], Moser [31], Gong [14, 15, 16], Stolovitch [36], etc. Bishop discovered an important geometry associated with M near an elliptic complex tangent p by proving the existence of a family of holomorphic disks attached to M shrinking down to p. He also proposed a problem concerning the uniqueness and smooth regularity of the geometric object obtained by taking the union of all locally attached holomorphic disks. Along these lines of the Bishop problem, there are works done by many people, including Hunter, Wells, Bedford-Gaveau, Kenig-Webster, Moser-Webster, Huang-Krantz, etc. Bishop’s problem was finally solved in 1998 in any dimensional case: Theorem 3.1 [28, 29, 22]. Let M ⊂ Cn (n  2) be a smooth n-dimensional submanifold with an elliptic tangent p ∈ M (0  λ < 12 ). Then there exists a smooth family of disjoint embedded holomorphic disks

(the local holomorphic hull of M near p) is a with boundaries in M , converging to p, whose union M smooth Levi flat (n + 1)-dimensional manifold-with-boundary such that M is contained in its boundary.

4

Normal form of elliptic Bishop surfaces with a non-vanishing Bishop invariant

There is a fundamental difference between the normal form theory for CR manifolds and that for Bishop surfaces. The latter is more along the lines of many normalization problems in classical mechanics and dynamics. To make it more precise, we notice that when one constructs a normal form, there always comes out a naturally truncated system, as we will discuss in more detail later. In the CR case, the truncated homogeneous system is linear and decoupled. However, in the Bishop surface case, the truncated homogeneous equations are non-decoupled and non-linear. To explain this more explicitly, we suppose that the hypersurface in C2 v = |z|2 + Owt (3) is expected to transform into a normal form as follows: v = |z|2 +

∞ 

N (k) (z, z, u).

k=1

has been determined up to degree m − 1 and we want to construct the normal form Suppose that N at the level of degree m, without changing the property at level before m−1. We write the transformation as follows: ∞ ∞   z = z + f (j−1) (z, w), w = w + g (j) (z, w). (k)

j=3

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We then get an equation called the m-truncation equation: L := Im(g (m) (z, |z|2 ) − 2iz, f (m−1) (z, |z|2 )) = N (m) + R(f (2) , g (3) , . . . , f (m−2) , g (m−1) ). We want L = 0 to have a unique solution. Otherwise, f (m−1) , g (m) will get in non-linearly at the later truncation as unknowns. For instance, |f (m−1) |2 will appear at the degree (2m − 2)-truncation term. To have a unique solution for L = 0, initial normalizations jump in. Namely, we have to restrict the 2 classification to a smaller group Bihnor 0,cm (C , 0), where L = 0 has a unique solution. L is called the Chern-Moser operator. In the Bishop surface case, there is a similar operator. In the CR case, we have that 2 Bih0 (C2 , 0)/Bihnor 0,cm (C , 0) is of the finite dimension and thus we are in a situation ∞ = ∞. a finite number In the Bishop surface case, we are in the situation of 2 Bih0 (C2 , 0)/Bihnor 0,B (C , 0), 2 where Bihnor 0,B (C , 0) is the group which makes the corresponding (Chern-Moser) operator have a zero 2 kernel space. Bihnor 0,B (C , 0) depends on the Bishop invariant. For instance, when λ = 0, we have 2 Bihnor 0,B (C , 0) =



 f (z, w), g(z, w) : f (z, w) = z + z 2 fˆ(z, w),

g(z, w) = w + Owt (3), with fˆ is a holomorphic function .

Since 2 Bih0 (C2 , 0)/Bihnor 0,B (C , 0)

is of infinite dimension for any λ, we come to a situation of ∞ ∞ for the space of invariants, which intuitively could be anything. This is the fundamental difference between the normalization problem for Bishop surfaces and strongly pseudoconvex hypersurfaces. To avoid such a difficulty, in the Bishop surface case, we have to work with the Moser operator L in the general group Bih0 (C2 , 0) and thus the non-linearity for the equation can not be avoided. In a celebrated paper of Moser-Webster [32], for λ = 0, they discovered a very nice reduction to avoid this situation and made the problem into a normalization problem for a pair of involutions interwined by a conjugate operator. However, their theory does not cover the intrigued elliptic Bishop surfaces with λ = 0. We next state the theorem of Moser-Webster: Theorem 4.1 [34]. (i) For a non-exceptional (λ = 0, 1/2, ∞ and λν 2 − ν + λ = 0 has no root of unity in the variable ν) Bishop surface, there exists a formal transformation such that in the new coordinates (z, w = u + iv) ∈ C2 , the surface is defined by u = z z¯ + (λ + us )(z 2 + z¯2 ),

v = 0,

 ∈ {0, 1, −1},

s ∈ Z+ .

(4.1)

(ii) For the elliptic case with λ = 0, the above transformation is biholomorphic. As an immediate consequence of the Moser-Webster theory, we conclude that there are only two and a half invariants for the elliptic Bishop surfaces with λ = 0. Namely, in this setting, we have ∞ = 2.5. ∞

5

Normal form of elliptic Bishop surfaces with a vanishing Bishop invariant

In [31], Moser first carried out a study for λ = 0 by applying the transformation from the group similar 2 to Bihnor 0,CM (C , 0). Hence, he needed only to deal with a linear and decoupled system to get the following normalization.

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Theorem 5.1 [33]. normal form:

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(i) The Bishop surface with a vanishing Bishop invariant has the following pseudo   aj z j . w = z z¯ + z s + z¯s + 2Re

(5.1)

js+1

Here s is the simplest higher order invariant of M at a complex tangent with a vanishing Bishop invariant, which we call the Moser invariant. (ii) If s = ∞, then the transformation of the surface to its pseudo-normal form can be made to be convergent. The above pseudo-normal form is still subject to the action of an infinite dimensional group 2 2 Bih0 (C2 , 0)/Bihnor 0,B (C , 0) = Aut0 (M∞ : w = |z| ).

Here, 2 Bihnor 0,B (C , 0) =



 f (z, w), g(z, w) : f (z, w) = z + z 2 fˆ(z, w),

g(z, w) = w + Owt (3), with fˆ is a holomorphic function .

And Aut0 (M∞ ) consists of the following transformations: w = wa(w)a(w),

z  = a(w)

z − wb(w) 1 − b(w)z

with a(0) = 0, and a(w), b(w) arbitrary power series in w. (Or all are in the formal sense.) Motivated by the papers [32] and [31], Moser asked the following two problems: Question I. What is the analytic regularity of the holomorphic hull filled in by complex disks attached to the Bishop surface with a vanishing Bishop invariant? Question II. Do Bishop surfaces with a vanishing Bishop invariant have finitely many invariants? How to solve the equivalence problem for Bishop surfaces with a vanishing Bishop invariant? The first question was solved by Huang-Krantz in [23]. The second problem concerns the higher order invariants for Bishop surfaces with λ = 0. It was answered in a recent paper of the authors [24]. The way we constructed the normal form is to directly work with the mappings from Bih0 (C2 , 0). Thus, we are dealing a non-linear and non-decoupled system. It seems to us that this is the first work in the normalization of complex analysis and geometry where one deals with a non-linear and non-decoupled truncated system. We now state the main results from [24]: Theorem 5.2 [25]. Let M be a formal Bishop surface which has an elliptic complex tangent at 0 with its Bishop invariant l = 0 and its Moser invariant s  3 and s < ∞. Then M can be transformed into the following formal normal form: s w = z  z¯ + z s + z¯ + ϕ(z  ) + ϕ(z  ),

where ϕ(z  ) =

∞  s−1 

aks+j z ks+j .

k=1 j=2

Such a formal transform is unique up to a composition from the left with a rotation of the form (z  , w ) = Rθ (z  , w ) := (eiθ z  , w ), where θ is a constant with eisθ = 1. Namely, if there is another formal equivalence map (z  , w ) = F ∗ (z, w) with F ∗ (0) = 0 that maps M into the following formal normal form: s w = z  z¯ + z s + z¯ + ϕ∗ (z  ) + ϕ∗ (z  )

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with ϕ∗ (z  ) =

∞  s−1 

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.

k=1 j=2

Then F ∗ = Rθ ◦ F for a certain θ with e

√ −1θs

= 1 and aks+j = e

√ −1jθ ∗ aks+j .

Theorem 5.3 [25]. Let M1 and M2 be real analytic Bishop surfaces with λ = 0 and s = ∞ at 0. Suppose that M1 has a normal form s w = z z + z + z¯ + 2Re 

 ¯

s

 ∞  s−1

aks+j z

 ks+j

bks+j z

 ks+j

 ;

k=1 j=2

and suppose that M2 has a formal normal form s w = z z + z + z¯ + 2Re 

 ¯

s

 ∞  s−1

 .

k=1 j=2

Then (M1 , 0) is biholomorphic to (M2 , 0) if and only if there is a constant θ, with esθ √ aks+j = eθj −1 bks+j for any k  1 and j = 2, . . . , s − 1.

√ −1

= 1, such that

We will not go into the discussion of the complicated proof in [24]. We just mention that our construction of the normal forms is done by a completely new weighting system from what is used in the literature. Also, our convergence argument is different from what people used before in the following aspects: (1) It is different from what Moser-Webster, Moser, Gong, Stolotvich, et al., used, which is along the lines of small divisor, KAM, etc. (2) It is different from what people used in the CR setting, which is along the lines of applying the reflection principle argument and the Artin approximation (see, for example, [3, 32, 31], etc.). Our argument is based on the hyperbolic geometry from the holomorphic disks attached to M . There are several immediate applications of the above results: (1) It shows that there are infinitely many invariants in this setting. (2) It can be used to show that it is generically not equivalent to an algebraic surface. (For the CR case, see similar results in [13, 27, 42].)

6

Normal form of Bishop surfaces with hyperbolic Bishop invariants

The study on the normalization problem for the hyperbolic complex tangents is quite different from that for the elliptic complex tangents. For a Bishop surface M with a non-exceptional hyperbolic complex tangent, Moser-Webster [32] showed that it must be formally equivalent to the model Mλ,,s = {(z, w) : Rew = zz + (λ + (Rew)s )(z 2 + z 2 ), Imw = 0}, where s is a positive integer and  ∈ {±1, 0}. MoserWebster [32] and Gong [16] also constructed various examples showing that the formal process is divergent in general. A natural question that arises here is the following: Problem 6.1. Construct the modular space for germs of non-exceptional hyperbolic Bishop surfaces, which are formally equivalent to Mλ,,s . This problem is closely connected with the well-known modular space problem for germs of holomorphic self-maps of (C, 0) with the identity as their linear term (see [39, 10, 1] for the modular space of the real analytic space with parabolic complex tangents). Motivated by problems from classical dynamics, it is interesting to find a necessary and sufficient condition for a hyperbolic Bishop surface, which is formally equivalent to the quadric model, to be holomorphically equivalent to the quadric. By Moser-Webster [32], a non-exceptional Bishop surface is holomorphically equivalent to the quadratic if and only if a certain holomorphic mapping associated to the surface, which is of the form σ(z) = μz + O(2) with μ = λ2 and λ a root of γλ2 − λ + γ = 0, can

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be holomorphically linearized. When the complex tangent point is elliptic and λ = 0, the linearization can be done by the method developed for the area-preserving mappings. Notice that it was shown by R¨ ussman [35] that a formally linearizable area-preserving analytic mapping is indeed analytically 1 log μ satisfies the Bruno linearizable under the Bruno condition. This essentially implies that if α = 2πi condition, then M must be holomorphically equivalent to the quadric if it is formally equivalent to. See also [14] for the holomorphic equivalence property under the Diophantine condition. Gong in [16] showed that there indeed exist some Bishop surfaces near a hyperbolic point which are formally but not holomorphically equivalent to the quadratic. Then we have the the following natural question: Is the Bruno condition also sufficient for the Bishop surface to be holomorphically equivalent to the quadric if it is formally equivalent to? Notice that for the Siegel linearization problem, the Bruno condition is both necessary and sufficient, by the famous work of Yoccoz [40]. (See also [33] for a nice description of the proof.) It would be very interesting to see if one can adapt Yoccoz’s method to the Bishop case. This problem is communicated to us by Gong.

7

An algebraic family of Bishop surfaces

The above formal normal form obtained in [24] solves completely the equivalence problem for Bishop surfaces with a vanishing Bishop invariant. However, it remains to be a difficult open problem to answer whether the normal form for a real analytic surface obtained in [24] is convergent or not. Hence, it is left to be an open problem to describe the modular space of the Bishop surfaces with λ = 0. The normal form in [24] was achieved by a very complicated iteration procedure, which makes it difficult to write out the normal form explicitly even if the surfaces are simple. For two given Bishop surfaces which are not defined by their normal forms, it is not easy to tell whether they are equivalent or not. In this part of the paper, we shall study the equivalence problem for a family of algebraic Bishop surfaces with one parameter. We will compute precisely the normal forms [24] for such surfaces up to a certain order. The specified surfaces studied in this paper make it possible to simplify many computations needed in the general case in [24]. This, besides solving the equivalence problem for such surfaces, may also shed some light for our further work on the convergence problem. The result can be stated as follows: az s+1 ) are holomorTheorem 7.1. (i) When s  4, w = zz + 2Re(z s + az s+1 ) and w = zz + 2Re(z s + iθ isθ phically equivalent if and only if a = ae , where θ is a constant satisfying e = 1. az 4 ) (ii) When both a and a both are real and a = a, then w = zz +2Re(z 3 +az 4 ) and w = zz +2Re(z 3 + are not holomorphically equivalent.

8

The proof of Theorem 7.1

The surface under consideration is defined by w = zz + 2Re(z s + az s+1 ).

(8.1)

Define the weight of z, z and w to be 1, s − 1 and s, respectively. Let E(z, z¯) (respectively, f (z, w)) be a formal power series in (z, z¯) (respectively, in (z, w)) without the constant term. We say E(z, z) = owt (k) if E(tz, ts−1 z) = o(tk ) for t → 0. Similarly, we say f (z, w) = owt (k) if f (tz, ts w) = o(tk ) for t → 0. Proof of Theorem 7.1. (i) We first consider the case of s  4. Suppose that the surface defined by (8.1) has the following approximate normal form: w = z  z  + 2Re(z s + b1 z s+2 + b2 z s+3 ) + owt (s + 3), through the transformation of the form  z  = z + f (z, w), w = w + g(w),

f (z, w) = O(|z|2 + |w|), g(w) = O(|w|2 ),

g(w) = g(w).

(8.2)

(8.3)

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Notice that we need only to consider the terms of the weighted degree bounded by s + 3. Since the weights of z and z are 1 and s − 1, respectively, we only need to consider the terms of the weight less than or equal to (s + 3) − (s − 1) = 4 in z  , terms of the weight less than or equal to (s + 3) − 1 = s + 2 in z  , and terms of the weight less than or equal to s + 3 in w . Hence we can write z  , z  and w as follows: ⎧  s  2 3 4 ⎪ ⎪ ⎨ z = z + a1 (zz + z ) + a1 z + a2 z + a4 z + owt (4), z  = z + a1 (zz + z s + az s+1 ) + a1 z 2 + owt (s + 2), ⎪ ⎪ ⎩ w = w + O(|w|2 ).

(8.4)

Substituting (8.4) into (8.2) and collecting the terms of degree 3, we obtain z(a1 zz + a1 z 2 ) + z(a1 zz + a1 z 2 ) = 0, which gives a1 = −a. Now we divide the proof of (i) into two parts according to s > 4 or s = 4. (1) When s > 4, we can write ⎧  2 3 4 ⎪ ⎪ ⎨ z = z − a1 z + a2 z + a4 z + owt (4), z  = z + a1 (zz + z s + az s+1 ) + owt (s + 2), ⎪ ⎪ ⎩ w = w + o (2s − 1). wt

Substituting (8.5) into (8.2), we get zz + 2Re(z s + az s+1 ) = (z − a1 z 2 + a2 z 3 + a4 z 4 )(z + a1 (zz + z s + az s+1 )) + (z − a1 z 2 + a2 z 3 + a4 z 4 )s + (z + a1 (zz + z s + az s+1 ))s + b1 (z − a1 z 2 + a2 z 3 + a4 z 4 )s+2 + b2 z s+3 + owt (s + 3). With an immediate simplification, we obtain ((1 − s)a1 − a)z s+1 + (aa1 − a21 + sa2 + Cs2 a21 + b1 )z s+2 +(a2 − a21 )z 3 z + (−aa21 + a2 a1 + sa4 − 2Cs2 a1 a2 − Cs3 a31 − (s + 2)a1 b1 + b2 )z s+3 +(a2 a1 + a4 )z 4 z = owt (s + 3). Here we have set Cnk :=

n! k!·(n−k)! .

Hence we obtain

a = (1 − s)a1 , a2 = a21 , a4 = −a2 a1 = −a21 a1 = −a31 . And consequently, b1 = − aa1 + a21 − sa2 − Cs2 a21  2 2  2  a a a a s(s − 1) + =−a −s − 1−s 1−s 1−s 2 1−s 2 2 a s−s s = (−2(1 − s) + 2 − 2s − s2 + s) = a2 = a2 . 2(1 − s)2 2(1 − s)2 2(1 − s) b2 = aa21 − a2 a1 − sa4 + 2Cs2 a1 a2 + Cs3 a31 + (s + 2)a1 b1 2  3 3 3    a a a a =a − +s + s(s − 1) 1−s 1−s 1−s 1−s  3 a s s(s − 1)(s − 2) a · a2 + + (s + 2) 6 1−s 1 − s 2(1 − s)

(8.5)

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=

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a3 s(1 + s) 3 (2s − 2s3 ) = a . 6(1 − s)3 3(1 − s)2

az s+1 ) are holomorphically Now suppose that w = zz + 2Re(z s + az s+1 ) and w = zz + 2Re(z s + equivalent with s > 4. By Theorem 1.2 of [24], we have s s a2 =

a2 · e2iθ , 2(1 − s) 2(1 − s)

s(1 + s) 3 s(1 + s) 3 3iθ a = a ·e . 2 3(1 − s) 3(1 − s)2

a. Here θ is a constant satisfying eisθ = 1. Hence we obtain a = eiθ (2) For s = 4, define the weights of z, z, and w to be 1, 3 and 4, respectively. Write z  , z  and w as follows: ⎧  4 2 3 4 ⎪ ⎪ ⎨ z = z + a1 (zz + z ) − a1 z + a2 z + a4 z + owt (4), z  = z + a1 (zz + z 4 + az 5 ) − a1 z 2 + owt (6), ⎪ ⎪ ⎩ w = w + o (7). wt

(8.6)

Substituting the above to w = z  z  + 2Re(z 4 + b1 z 6 + b2 z 7 ) + owt (7), we obtain   zz + 2Re(z 4 + az 5 ) = z + a1 (zz + z 4 ) − a1 z 2 + a2 z 3 + a4 z 4   4  × z + a1 (zz + z 4 + az 5 ) − a1 z 2 + z + a1 (zz + z 4 ) − a1 z 2 + a2 z 3 + a4 z 4 + b1 (z + a1 (zz + z 4 ) − a1 z 2 + a2 z 3 + a4 z 4 )6 + b2 z 7 + owt (7). An immediate simplification shows that (−a − 3a1 )z 5 + (a1 a − a21 + 4a2 + 6a21 + b1 )z 6 + (a2 − a21 )z 3 z +(−a21 a + a2 a1 + 4(a1 + a4 ) − 12a1 a2 − 4a31 − 6b1 a1 + b2 )z 7 + (a1 a2 + a1 + a4 + 4a1 )zz 4 = owt (7). Now it is easy to see that a = −3a1 ,

a2 = a21 =

a2 , 9

a4 = −a1 a2 − 5a1 =

a a2 · −5· 3 9

 −

a 3

 =

a3 5 + a. 27 3

And consequently, b1 = −a1 a + a21 − 4a2 − 6a21 =

a2 a2 a2 2 a2 + −4· −6· = − a2 , 3 9 9 9 3

b2 = a21 a − a2 a1 − 4(a1 + a4 ) + 12a1 a2 + 4a31 + 6b1 a1     a a3 a2 a 5 a3 − − −4 − + + a = 9 3 327 3  3 2    9 a a 2 2 a a · +4· − +6· − a · − +12 · − 3 9 27 3 3   1 4 12 4 12 3 16 20 3 16 1 + − − − + a − a= a − a. = 9 27 27 27 27 9 3 27 3 az 5 ) are holomorphically equivalent, then by If w = zz + 2Re(z 4 + az 5 ) and w = zz + 2Re(z 4 + Theorem 1.2 of [24] as above, we get   2 2 2 2iθ 20 3 16 20 3 16 − a2 = − a ·e , a − a=

a − a · e3iθ . 3 3 27 3 27 3 aeiθ or a = − aeiθ . When Here θ is a constant satisfying e4iθ = 1. By the first equation, we get a = iθ iϑ 4iϑ ae with ϑ = θ + π. Then we have e = 1. This completes the proof of a = − ae , we can write a = Theorem 7.1(i).

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(ii) Now we assume that s = 3 and a = a. The surface is then defined by w = zz + 2Re(z 3 + az 4 ). Set the weights of z, z and w to be 1, 2 and 3, respectively. Suppose that the above surface has the following approximate normal form: 3 5 8 w = z  z¯ + z  + z¯ 3 + bz  + ¯bz¯ 5 + cz  + c¯z¯ 8 + owt (8),

(8.7)

with the transformation ⎧ 1 1 ⎪  ⎪ ¯w + az 2 + a1 z 3 + a2 zw + a3 z 4 + a4 z 2 w + a5 w2 + a6 z 5 + a7 z 3 w ⎪ ⎨ z =z − 2a 2 2 6 4 7 zw + a + a 8 9 z + a10 z w + a11 z + owt (7), ⎪ ⎪ ⎪ ⎩ w = w + b w2 + o (8). 2

wt

Substituting the above to (8.7), repeating the procedure in the proof of Theorem 3.1 of [24] and comparing the coefficients of z 4 , zz, z 2 z 2 , z 3 z, z 5 , zz 4 , z 2 z 3 , z 6 , zz 5 , z 7 , z 8 , respectively, we can get 1 a1 = 2¯ a + a2 , 4 1 3 1 1 a− a ¯ , a2 = − a3 + a¯ 3 2 6 5 a3 = a3 + 3a¯ a, 8 1 3 2 13 4 41 2 159 2 15 a − a a a ¯ − a− a ¯ a , a4 = ¯− 192 16 16 8 12 29 4 1 3 2 39 2 135 2 27 a ¯ + a a ¯ a+ a + a ¯, a5 = − ¯ + a 192 6 16 16 8 19 4 51 2 129 2 3 a6 = a + a a a ¯ + a, ¯+ 32 8 8 4 29 4 279 2 315 2 209 3 1 3 2 29 5 a7 = −15¯ ¯ − a − a ¯ a− a a ¯ a + a , a+ a ¯− a 48 8 16 48 24 192 231 4 1209 2 291 223 3 1 3 2 49 5 a ¯− a ¯ + a + 69¯ a a ¯ a − a . a9 = a2 a + ¯− a 8 64 16 16 4 64 and

3 1 1 3 3 9 ¯ − a2 , b 2 = − a3 − a ¯ + a¯ a, b=− a 2 4 2 2 2 231 4 1179 2 459 2 297 1 3 2 a ¯+ a ¯ − a − a ¯ a+ a ¯ a + 4a5 . c=− 4 32 8 8 2 For convenience, we set c(a) = t1 a + t2 a4 + t3 a2 + t4 a2 a + 12 a3 a2 + 4a5 , that is, we define t1 = − 297 4 , t2 = 231 1179 459 , t = − , t = − . 3 4 32 8 8 az 4 ) are holomorphically equivalent, then Now if w = zz + 2Re(z 3 + az 4 ) and w = zz + 2Re(z 3 +   3 2 9 ¯ 3 2 2iθ 9 ¯− a = − a− a e , (8.8) − a 2 4 2 4   4 2 1 1 3 2 a + t2 a + t3 a 2 + t4 a a+ a a + 4 a5 e2iθ . (8.9) t1 a + t2 a4 + t3 a2 + t4 a2 a + a3 a2 + 4a5 = t1 2 2 Here θ is a constant satisfying e3iθ = 1. Set x = a − aeiθ , y = a + aeiθ . Then (8.8) becomes x = − 61 xy. If x = 0, we have a = aeiθ . When x = 0, then xx = − 61 y. Hence we can write that x = reiϑ (r > 0, ϑ ∈ −2iϑ . Substituting them to (8.9), we obtain R), y = −6e t1 re−iϑ − 3t2 reiϑ (r2 e−2iϑ + 62 e4iϑ ) − 6rt3 e−iϑ 2   −iϑ  − 6e2iϑ re + t4 reiϑ − 3reiϑ (−reiϑ − 6e−2iϑ ) 2  3  2   −iϑ  − 6e2iϑ 1 −reiϑ − 6e−2iϑ 1 3 −3iϑ −iϑ re 3iϑ − 6re 27re + r e + + 2 2 2 4

698

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 +4

reiϑ − 6e−2iϑ 2

5

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−reiϑ − 6e−2iϑ 2

= 0.

(8.10)

However, it is not easy to judge the solvability of (8.10) for r > 0, ϑ ∈ R. Instead, we shall prove that if a, a are real and a = a, then (8.10) does not have a solution. Indeed, in (8.8) and (8.9), we now have eiθ = 1. Hence (8.8) becomes a + a = −6 and (8.9) is simplified as  t1 (a − a) + t2 (a − a ) + t3 (a − a ) + t4 (a − a )+ 4

4

2

2

3

3

 1 + 4 (a5 − a5 ) = 0. 2

Deleting the common factor of (a − a) in both sides, we obtain 9 t1 + t2 (a + a)(a2 + a2 ) + t3 (a + a) + t4 (a2 + a a+ a2 ) + (a4 + a3 a + a2 a2 + a a3 + a4 ) = 0. 2

(8.11)

Substituting a + a = −6 into (8.11), we get 9 a) − 6t3 + t4 (62 − a a) + (64 − 3 · 62 a a + a2 a2 ) = 0. t1 − 6t2 (62 − 2a 2 A simple computation shows     9 2 2 9 9 a a + t1 − 63 t2 − 6t3 + 62 t4 + · 64 = 0. a + 12t2 − t4 − · 3 · 62 a 2 2 2 Notice that 9 231 459 9 · 3 · 62 = 12 · + − · 3 · 62 = −342, 2 32 8 2 1179 9 297 231 459 9 4 12069 − 63 · +6· − 62 · + ·6 = . t1 − 63 t2 − 6t3 + 62 t4 + · 64 = − 2 4 32 8 8 2 4 12t2 − t4 −

Hence 92 (a a)2 − 342a a+

12069 4

= 0. The equation has the following solution:

a a=

342 ±

 3422 − 4 ·

9 2

·

12069 4

9

.

Recall that we already have a + a = −6. By (a + a)2  4a a, we obtain

4·

342 −

 3422 − 4 ·

9 2

·

12069 4

9

which gives 3422 −

9 · 12069 = 62653.5  68121 = 2

 36,

 2 9 · 36 342 − . 4

This is a contradiction. This completes the proof of Theorem 7.1.

Acknowledgements The first author was supported in part by US National Science Foundation (Grant No. 0801056); the second author was supported in part by National Natural Science Foundation of China (Grant No. 10901123), Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20090141120010), Ky and Yu-Fen Fan Fund from American Mathematical Society, and a research fund from Wuhan University (Grant No. 1082002). The authors appreciate very much Professor X. Gong for his discussions on a normalization problem for the hyperbolic complex tangents. They also thank Professor S. Ji for several related discussions.

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