Projection on Segre varieties and determination of holomorphic mappings between real submanifolds Dedicated to Professor Sheng GONG on the occasion of his 75th birthday

It is shown that a germ of a holomorphic mapping sending a real-analytic generic submanifold of finite type into another is determined by its projection on the Segre variety of the target manifold. A necessary and sufficient condition is given for a germ of a mapping into the Segre variety of the target manifold to be the projection of a holomorphic mapping sending the source manifold into the target. An application to the biholomorphic equivalence problem is also given.     

Projection on Segre varieties and determination of holomorphic mappings between real submanifolds

It is shown that a germ of a holomorphic mapping sending a real-analytic generic submanifold of finite type into another is determined by its projection on the Segre variety of the target manifold. A necessary and sufficient condition is given for a germ of a mapping into the Segre variety of the target manifold to be the projection of a holomorphic mapping sending the source manifold into the target. An application to the biholomorphic equivalence problem is also given. Dedicated to Professor Sheng GONG on the occasion of his 75th birthday     

THE GROWTH THEOREM FOR STARLIKE MAPPINGS ON BOUNDED STARLIKE CIRCULAR DOMAINS

1.IntroductionOnthegeometricfunctiontheoryofonecomplexvariable,thefollowinggrowthandi-coveringtheoremiswellknown(see[2]).TheoremA.Foreachno~alizedunivalentjunctionfontheunitdiscDCC,ESPecially,theleft-handsideOftheaboveinequalityimpliesf(D)2ID.ForeachzED,z/0,equalityoccursintheaboveinequalityifandonlyiffisKoe6efunctionK(z)=theoritsrotatione--"K(e"z).Itisnaturaltoextendthisandotherresultsonthegeometricfunctiontheoryofonevariablestoseveralvariables.Butasearlyasfiftyyearsago,H.Cartanpointe…     

ON THE SHARP GROWTH, COVERING THEOREMS FOR NORMALIZED BIHOLOMORPHIC MAPPINGS IN C^n

In this article.a normalized biholomorphic mapping f defined on bounded starlike circular domain in C″is considered,where z=0 is a zero of order k 1 of f(z)(?) The sharp growth.covering theorems for almost starlike mappings of order a and starlile mappings of order wave established.Meanwhile,the construction of the above mappings on bounded starlike circular domain in C~n is also discussed,it provides the extremal mappings for the growth,covering theorems of the above mappings.     

Convergence and finite determination of formal CR mappings

It is shown that a formal mapping between two real-analytic hypersurfaces in complex space is convergent provided that neither hypersurface contains a nontrivial holomorphic variety. For higher codimensional generic submanifolds, convergence is proved e.g. under the assumption that the source is of finite type, the target does not contain a nontrivial holomorphic variety, and the mapping is finite. Finite determination (by jets of a predetermined order) of formal mappings between smooth generic submanifolds is also established.

     

Growth theorem of convex mappings on bounded convex circular domains

The growth theorem and the 1/2-covering theorem are obtained for the class of normalized biholomorphic convex mappings on bounded convex circular domains, which extend the corresponding results of Sufridge, Thomas, Liu, Gong, Yu, and Wang. The approach is new, which does not appeal to the automorphisms of the domains; and the domains discussed are rather general on which convex mappings can be studied, since the domain may not have a convex mapping if it is not convex. Project supported by the National Natural Science Foundation of China and the State Education Commission Doctoral Foundation.     

关于C.Fefferman定理与全纯映射的边界扩充

C．Fefferman定理证明了光滑有界强拟凸域之间的双全纯映射可以光滑延拓到边界，这个结果已经被推广到各种情形．其中Bell和Catlin以及Diederich和Fornaess独立地将其推广到拟凸域的逆紧全纯映射．本文较全面地综述了C．Fefferman定理的推广情况以及Bergman投射的边界正则性问题，同时对如何去掉Bell和Catlin以及Diederich和Fornaess定理条件中的拟凸性给出一个新观察，提出一个解决方向并且说明在具体情况下这个新观察确实是可以提供答案的．     

Reinhardt域B_p上正规化双全纯完全准凸映射的齐次展式

本文主要研究C~n中有界凸Reinhardt域B_p上正规化双全纯完全准凸映射的齐次展开式问题,设f(z):B_p→C~n(p＞2)是一个正规化双全纯完全准凸映射,k是满足k＜p≤k+1的自然数,那么f_j(z)=z_j+α_(j2)z_j~2+…+α_(jk)z_j~k+O(|z|~(K+1)),其中f_j是f的第j个分量,j=1,2…．,n．     

A local version of Wong-Rosay's theorem for proper holomorphic mappings

In the present paper, we generalize Wong-Rosay's theorem for proper holomorphic mappings with bounded multiplicity. As an application, we prove the non-existence of a proper holomorphic mapping from a bounded, homogenous domain in onto a domain in whose boundary contains strongly pseudoconvex points.

     

Théorème de Poincaré–Alexander pour les domaines modèles

The Poincaré–Alexander Theorem states that holomorphic mappings defined on an open subset of the unit ball of ${\mathbb{C}}^n$ may, under certain conditions, be extended to a biholomorphism of the unit ball. In a complex manifold, every strongly pseudoconvex homogeneous domain is biholomorphic to the unit ball. In an almost complex manifold, the unit ball is not the only strongly pseudoconvex homogeneous domain. A strongly pseudoconvex homogeneous domain is biholomorphic to a model domain. The aim of this paper is to extend this theorem to model domains.     

B~n上推广的Roper-Suffridge算子的性质

讨论改进后的Roper-Suffridge延拓算子保持双全纯映照子族的性质.借助双全纯映照子族的解析特征及其偏差结论,得到改进后的Roper-Suffridge延拓算子在一定条件下保持α次殆β型螺形映照、α次β型螺形映照、强β型螺形映照的性质,从而得到改进后的算子在一定条件下保持α次殆星形性、α次星形性和强星形性.所得结论为在多复变数空间中构造这些双全纯映照提供了一种新的途径.     

第一类典型域上的Bloch常数

引入第一类典型域R_I(m,n)上的全纯映照子族H_k(R_I(m,n)),当k→+∞时,该映照族就是R_I(m,n)上的局部双全纯映照族.建立了H_k(R_I(m,n))上的Bonk偏差定理.当k=1和k→+∞时,其结果分别都回到了FitzGerad和龚升关于典型域R_I(m,n)上的Bonk偏差定理.当m=n=1时,其结果又回到了Liu和Minda在单位圆盘上的偏差定理.应用偏差定理,给出了映照族H_k(R_I(m,n))上的Bloch常数估计,其结果补全了从k=1和k→+∞之间的R_I(m,n)上Bloch常数估计的所有结果,而且把单位球上的Bloch常数估计推广到R_I(m,n)上.     

Linear cycle spaces in flag domains

Let Z = G/Q, a complex flag manifold, where G is a complex semisimple Lie group and Q is a parabolic subgroup. Fix a real form and consider the linear cycle spaces , spaces of maximal compact linear subvarieties of open orbits . In general is a Stein manifold. Here the exact structure of is worked out when is a classical group that corresponds to a bounded symmetric domain B. In that case is biholomorphic to B if a certain double fibration is holomorphic, is biholomorphic to otherwise. There are also a number of structural results that do not require to be classical. 19 March 1999     

一类多复变全纯映照子族的增长和偏差定理

在一般复Banach空间X中的单位球B上引入一类全纯映照族M_g.考虑B上满足条件(Df(x))~(-1)f(x)∈M_g的正规化局部双全纯映照f(x)(其中x=0是f(x)-x的k+1阶零点)并得到其增长定理.作为应用,也得到了C~n中单位多圆柱D~n上映照f关于Jacobi矩阵Jf(z)的偏差定理,该结果统一和推广了星形映照许多子族的相应结论.     

A classification of proper holomorphic mappings between generalized pseudoellipsoids of different dimensions

ABSTRACT

We classify proper holomorphic mappings between generalized pseudoellipsoids of different dimensions. Those domains are parametrized by the exponents. The relations among them are also obtained. Main tool is the orthogonal decomposition of a CR bundle. Such a decomposition derives the ‘variable-splitting’ of the mapping.     

Unbounded domains in ℂ2 with non-compact automorphisms group

Let D be a domain in ?² such that the orbit of an internal point under the group Aut(D) of the holomorphic automorphisms of D accumulates to a smooth boundary point p. It is well known that if p is a strictly pseudoconvex point, such a local information forces D to be biholomorphic to the ball. We prove that if p is only weakly pseudoconvex of finite type, the domain D need not to be biholomorphic to a natural standard model. On the other end our results show that, under some convexity assumptions, D may be realized as a bounded domain with real analytic boundary except at most at one point.     

Three Generic Results in Holomorphic Fixed Point Theory

We establish three theorems which show that most of the bounded holomorphic self-mappings of a star-shaped domain in a complex Banach space map it strictly inside itself. According to the Earle–Hamilton fixed point theorem, each such mapping has a unique fixed point.     

Localization principle of automorphisms on generalized pseudoellipsoids

We deduce global biholomorphic equivalence from local biholomorphic equivalence near boundary points in the class of generalized pseudoellipsoids, extending local biholomorphisms to global ones. We discuss related results on holomorphic proper maps.     

Proper holomorphic maps on an irreducible bounded symmetric domain of classical type

In this paper it is extended a result by W. Rudin on proper holomorphic maps from the open ball ofC ⁿ intoC ⁿ. I show that, forn>1, every proper map from an irreducible bounded symmetric domain of classical type intoC ⁿ, can be obtained, module biholomorphic maps, via a finite reflection group.     

Bloch Constant of Holomorphic Mappings on the Unit Ball of ℂ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>*

In this paper, the authors establish distortion theorems for various subfamilies Hk(B) of holomorphic mappings defined in the unit ball in Cn with critical points, where k is any positive integer. In particular, the distortion theorem for locally biholomorphic mappings is obtained when k tends to oo. These distortion theorems give lower bound son det f(z) and Redet f'(z). As an application of these distortion theorems, the authors give lower and upper bounds of Bloch constants for the subfamilies βk (M) of holomorphic mappings. Moreover, these distortion theorems are sharp. When B is the unit disk in C, these theorems reduce to the results of Liu and Minda. A new distortion result of Re det f'(z) for locally biholomorphic mappings is also obtained.