Hongjun LIChunhui QIUSchool of Mathematical Sciences,Xiamen University,Xiamen 361005,China E-mail:henanlihj@126.com;chqiu@xmu.edu.cn

THE HOLOMORPHIC AUTOMORPHISM GROUP OF HIGHER DIMENSION THULLEN DOMAIN∗

In this paper,we give the holomorphic automorphism group of the higherdimensional generalization of Thullen domain.

holomorphic automorphism group;isotropic subgroup;Thullen domain

2010 MR Subject Classification32A10

1　Introduction

In 1931,Thullen[1]succeeded in classifying the two dimensional Reinhardt domains,and gave the holomorphic automorphism group of Thullen domain

The holomorphic automorphism group on Dpis

where o≤θ,ψ<2π,α∈ℂ,|α|<1.In 1932,Cartan[2]got result(1.2)again by the Lie group technique.

In 1968,Naruki[3]investigated the equivalence problem for a special kind of Reinhardt domains including the higher-dimensional generalization of Thullen domain

where p=(p1,···,pn2),pk>o,pk/=1,1≤k≤n2and different from two.He also indicated a more wider class of domains defined as follows.For α=(α1,···,αs)and N=(n1,···,ns) (where αj>o(1≤j≤s),nj(1≤j≤s):positive integers),define the domain as follows:

where(z11,···,z1n1,z21,···,z2n2,···,zs1,···,zsns)is the coordinate of ℂnHe solved the equivalence problem.But the Lie algebra of the holomorphic automorphism group determined in his paper was not clear,he also did not give the holomorphic automorphism group.In the late 197os,Toshikazu[4]and Webster[5]generalized these works and further classified bounded Reinhardt domains in general case.

A bounded domain D is called a Reinhardt domain if it is always mapped onto itself by any of the following transformations:

where o≤θ1,θ2,···,θn<2π.

We denote a class of Reinhardt domain by

where 1≤ro,but p/=2.The domain defined by(1.6)is the higher-dimensional generalization of Thullen domain.We call(1.6)higher dimension Thullen domain.

Then(1.6)can be written by

Let D be a bounded domain in ℂn,and the origin o∈D.Denote by Aut(D)the holomorphic automorphism group of the domain D,Aut(D)othe unit connected component of Aut(D), and Iso(D)the isotropic subgroup with the origin of Aut(D)and Iso(D)othe unit connected component of Iso(D).

Cartan[6]proved that Aut(D)is a real Lie group and Iso(D)is a compact subgroup of Aut(D).In this paper,L is called the Lie algebra of a Lie group G,if L consists of all left invariant vector fields of the Lie group G.Moreover,aut(D)and iso(D)denote the Lie algebra of the Lie group Aut(D)and Iso(D),respectively.

For a bounded Reinhardt domain D containing the origin in ℂn,take a fixed index j(1≤j≤n),then Iso(D)ohas a 1-dimension real Lie subgroup

In this paper,we firstly determine the Lie algebrathen determine

First,we introduce some results of Cartan[6].

Lemma 1.1(see[6])Let D be a bounded domain in ℂn,and the origin o∈D.Suppose that the Taylor expansions at z=o of X=ξ(z)

respectively,where ξ(z)=(ξ1(z),···,ξn(z)),η(z)=(η1(z),···,ηn(z)),

Then X=Y if and only if A=B,where A,B∈gl(n,ℂ).In particular,X=o if and only if A=o.

Lemma 1.2(see[6])Let D be a bounded domain in ℂn,and the origin o∈D.Given a vector field∈aut(D).Then X,−1X∈aut(D)if and only if X=o;X∈iso(D) if and only if ξ(o)=o.

2　The Isotropic Subgroup with the Origin on

In this section,we determine the Lie algebra iso

consist of all vector fields such as the following forms

where Pk(z)=(P1k(z),···,Pnk(z))and Pjk(z)are homogeneous polynomials of degree k for 1≤j≤n,k≥o.In fact,by Lemma 1.2,we know Po(z)=o.By a direct calculation,we have

Take l/=m,1≤l≤n,by a direct calculation,we have

Next we prove Re(bjj)=o,1≤j≤n.In order to calculate exp(tXo),t∈ℝ,we solve the complex ordinary differential equation systems

with the initial values uj(o)=zj,1≤j≤n.The solutions arethat is,

For the sake of convenience,we firstly discuss the case of A12.We can calculate

By Lemma 1.2,we have b12=o.For the same reason,if b12=o,then b21=o.Note that forThen

Hence b12∈ℝ.The one-parameter subgroup w(t)=exp(tA12),∀t∈ℝ is a unique holomorphic solution of the complex ordinary differential equation systems

with the initial values wk(o)=zkfor 1≤k≤n.By the theory of ordinary differential equation systems,we have

By Xu[8],the characteristic boundaryand any holomorphic automorphism ofis also a holomorphic automorphism of the closed domain

On the other hand,take(z1,z2,z3···,zn)=(o,ζ,o,···,o),|ζ|=1,then

Hence b12=−1,that is,

Analogously,when 1≤l

If Alm/=o,then suppose AlmWe solve the complex ordinary differential equation systems of the one-parameter subgroup w(t)=exp(tAlm):

with the initial values wk(o)=zkfor 1≤k≤n.The solutions are,we get blm=−1,that is,1=|cost|2+|sint|p,∀t∈ℝ,then p=2.But on the other hand p/=2,it is a contradiction.

Immediately,we obtain the following theorem.

ProofBy Section 3 in[3],we know that Isois a simple connected Lie group,that is,

Denote Sn

First we prove Unwhere Unis the unitary group which is the set of all n×n unitary matrices.In fact,take any

On the contrary,take any U∈Un,then there exists V∈Un,such that

holomorphic solution of the complex ordinary differential equation systems

Thus Iso is constructed by the transformation as follows

The theorem is completely proved.

3　The Holomorphic Automorphism Group on

where 1≤k≤r.

ProofCartan[7]proved a homomorphic function defined on any bounded circular domain could be expanded the uniformly convergent homogeneous polynomial power series,so any

where ξjk(z)are homogeneous polynomials on z of degree k for 1≤j≤n,k≥o.

By Lemma 1.1,we have Z4+Z2=o,so ξjk(z)=o,1≤j≤n,k≥3.Thus we have

where at least one ξjo(o)is nonzero for 1≤j≤n.Then

and when l/=k,

Therefore Xlk∈iso,and by Lemma 1.1,Xlk=o,where l/=k.Immediately,we get

Note that if bk=o,then Xk=Yk=o.

If bk/=o,by(3.7)and(3.8),for any ζ∈ℂ,we have

Hence,we can suppose bk=1.So we haveThe one-parameter subgroup w(t)=exp(tXk),∀t∈ℝ is a unique holomorphic solution of the complex ordinary differential equation systems

with the initial values wl(o)=zlfor 1≤l≤n.t∈ℝ,it is a contraction that/=o.The solutions are

where t∈ℝ.Similar to the proof of Lemma 2.1,we can get

Next,we determine Xkand Yk,where 1≤k≤n.

For 1≤k≤r,suppose bk/=o.By(2.8),take(z1,···,zk,···,zn)=(o,···,ζ,···,o), |ζ|=1,then

For r+1≤k≤n,suppose bk/=o.By(2.8),take(z1,···,zk,···,zn)=(o,···,ζ,···,o), |ζ|=1,then

Immediately,we obtain the following theorem.

where a∈ℂ,α,β∈ℂr,D is a n×n complex matrix,is the traversal of the Lorentz group of type(1,r),i.e., and V∈Un−r.

ProofBy Section 3 in[3],we know that Autis a simple connected Lie group,that is,

The one-parameter subgroup w(t)=exp(t(cosθkXk+sinθkYk)),t∈ℝ is a unique holomorphic solution of the complex ordinary differential equation systems

with the initial values w(o)=z,where o≤θk<2π.The solutions are

where a∈ℂ,α,β∈ℂr,D is a r×r complex matrixis the traversal of the Lorentz group of type(1,r),i.e., and V∈Un−r.

The theorem is completely proved.

RemarkThe process of this paper can be generalized to more general case(1.4).

AcknowledgementsThe authors would also like to thank Prof.Yichao Xu for giving us useful discussions.

References

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[4]Sunada T.Holomorphic equivalence problem for bounded Reinhardt domains.Math Ann,1978,235(2): 111–128

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[9]Xu Y.On the classification of symetric Schlicht domains in several complex variable.Shuxue Jinzhan,1965, 8(2):109–144(Chinese)

∗May 18,2015;†

Chunhui QIU.

December 14,2015.Supported by the National Natural Science Foundation of China(11571288,11271304,11171277).