On the automorphisms of circular and Reinhardt domains in complex banach spaces

The group Aut(D) of all biholomorphic automorphisms of a bounded circular domain D in a complex Banach space E is discussed. As an application the group Aut[D] is computed for the open unit balls of certain classical Banach spaces.     

On Biholomorphy in Infinite Dimensions

In this article, we consider the question of when a holomorphic mapping on a domain in a complex Hilbert space into itself has a holomorphic inverse. We give a strengthened version of a known result that involves a Fredholm condition on the mapping. We show that holomorphic mappings on certain domains that are biholomorphic near the boundary are biholomorphic on the domain itself.      

On the existence of complete Kähler metrics of negative Riemannian curvature bounded away from zero on ellipsoidal domains inC n

Summary The purpose of this paper is to prove that every ellipsoidal domain in Cⁿ admits a complete Kähler metric whose Riemannian sectional curvature is bounded from above by a negative constant (Theorem 1). We construct a Kähler metric, in a natural way, as potential of a suitable function defining the boundary (§2). Directly we compute the curvature tensor and we find upper and lower bounds for the holomorphic sectional curvature (§ 4, § 5). In order to prove the boundness of Riemannian sectional curvature we use finally a classical pinching argument (§ 6). We also obtain that for certain ellipsoidal domains the curvature tensor is very strongly negative in the sense of [15] (§ 3). Finally we prove that the metric constructed on ellipsoidal domains in Cⁿ is the Bergman metric if and only if the domain is biholomorphic to the ball (Theorem 2). In [8], [9] R. E. Greene and S. G. Krantz gave large families of examples of complete Kähler manifolds with Riemannian sectional curvature bounded from above by a negative constant; they are sufficiently small deformations of the ball in Cⁿ, with the Bergman metric. Before the only known example of complete simply-connected Kähler manifold with Riemannian sectional curvature upper bounded by a negative constant, not biholomorphic to the ball, was the surface constructed by G. D. Mostow and Y. T. Siu in [14], to the best of the author's knowledge, is not known at present if this example is biholomorphic to a domain in Cⁿ.     

Biholomorphic isomorphisms between Teichmüller curves

We determine the biholomorphic isomorphisms between Teichmüller curves for torsion free Fuchsian groups. We show that, except in some special cases, a biholomorphic isomorphism between two Teichmüller curves is an allowable mapping; in particular, a biholomorphic automorphism of a Teichmüller curve is induced by an element of the modular group. As a by-product of the proof, we also obtain some topological results on self-mappings of surfaces. Research supported by the National Natural Science Foundation of China.     

Orbits of triples in the Shilov boundary of a bounded symmetric domain

Let be a bounded symmetric domain of tube-type, S its Shilov boundary, and G the neutral component of its group of biholomorphic transforms. We classify the orbits of G in the set      

The Lempert Theorem and the Tetrablock

In this paper we show that the Lempert property (i.e., the equality between the Lempert function and the Carathéodory distance) holds in the tetrablock, a bounded hyperconvex domain which is not biholomorphic to a convex domain. The question whether such an equality holds was posed by Abouhajar et al. in J. Geom. Anal. 17(4), 717–750 (2007).     

Fiber spaces over Teichmüller spaces for Fuchsian groups with elliptic elements

We determine the biholomorphic fiber preserving isomorphisms of fiber spaces over Teichmüller spaces for Fuchsian groups with elliptic elements. We show that except in some special cases a biholomorphic fiber preserving isomorphism between two Bers fiber spaces is always an allowable mapping. We find that the situation is different for Teichmüller curves, showing that in general there are some other biholomorphic fiber preserving isomorphisms between Teichmüller curves besides the allowable mappings. Research supported by the National Natural Science Foundation of China.     

Classification of sequentially weakly continuous $JB^*$-triples

Let D be the open unit ball of a -triple A and let Aut(D) be the group of all biholomorphic automorphisms of D. It is shown that every element of Aut(D) is sequentially weakly continuous if and only if every primitive ideal of A is a maximal closed ideal and is a type I -triple without infinite-spin part. Implications for general structure theory are explored. In particular, it is deduced that every -triple A contains a smallest ideal J for which the sequentially weakly continuous biholomorphic automorphisms of the open unit ball of A/J are all linear. Received August 27, 1998; in final form February 10, 1999     

Finite p-groups all of whose proper subgroups have small derived subgroups

Let G be a finite p-group.If the order of the derived subgroup of each proper subgroup of G divides pi,G is called a Di-group.In this paper,we give a characterization of all D1-groups.This is an answer to a question introduced by Berkovich.     

Criteria for biholomorphic convex mappings on the unit ball in Hilbert spaces

In this paper, we give a necessary and sufficient condition that a locally biholomorphic mapping f on the unit ball B in a complex Hilbert space X is a biholomorphic convex mapping, which improves some results of Hamada and Kohr and solves the problem which is posed by Graham and Kohr. From this, we derive some sufficient conditions for biholomorphic convex mapping. We also introduce a linear operator in purpose to construct some concrete examples of biholomorphic convex mappings on B in Hilbert spaces. Moreover, we give some examples of biholomorphic convex mappings on B in Hilbert spaces.     

Locally biholomorphic mappings of multiconnected domains

Sufficient conditions were found in [1, 2] for a domain in the complex plane to admit a finitely valent locally biholomorphic mapping of the whole plane. In this article we find the relevant necessary and sufficient conditions. Also, we answer the question that was posed by Aksent’ev and Ul’yanov in connection with the problem under consideration. The answer yields a lower bound for the valency while generalizing a result of [3] to polydisks.     

Aspherical Kähler manifolds with solvable fundamental group

We survey recent developments which led to the proof of the Benson-Gordon conjecture on Kähler quotients of solvable Lie groups. In addition, we prove that the Albanese morphism of a Kähler manifold which is a homotopy torus is a biholomorphic map. The latter result then implies the classification of compact aspherical Kähler manifolds with (virtually) solvable fundamental group up to biholomorphic equivalence. They are all biholomorphic to complex manifolds which are obtained as a quotient of $\mathbb{C}^{n}We survey recent developments which led to the proof of the Benson-Gordon conjecture on K?hler quotients of solvable Lie groups. In addition, we prove that the Albanese morphism of a K?hler manifold which is a homotopy torus is a biholomorphic map. The latter result then implies the classification of compact aspherical K?hler manifolds with (virtually) solvable fundamental group up to biholomorphic equivalence. They are all biholomorphic to complex manifolds which are obtained as a quotient of \mathbbCⁿ\mathbb{C}^{n} by a discrete group of complex isometries.     

Extension operators via semigroups

The Roper-Suffridge extension operator and its modifications are powerful tools to construct biholomorphic mappings with special geometric properties. The first purpose of this paper is to analyze common properties of different extension operators and to define an extension operator for biholomorphic mappings on the open unit ball of an arbitrary complex Banach space. The second purpose is to study extension operators for starlike, spirallike and convex in one direction mappings. In particular, we show that the extension of each spirallike mapping is A-spirallike for a variety of linear operators A. Our approach is based on a connection of special classes of biholomorphic mappings defined on the open unit ball of a complex Banach space with semigroups acting on this ball.     

有界凸平衡域上的双全纯凸映照的判别准则

本文讨论Cn中有界强凸平衡域和凸平衡域上局部双全纯映照成为双全纯凸映照的充要条件,从而得到Reinhardt域Dp= 上双全纯凸映照的充要条件,其中Pj≥2(j=1,2,…,n).     

Distortion theorems for subclasses of starlike mappings along a unit direction in ℂ

In this paper, we obtain a distortion theorem of Jacobian matrix for biholomorphic subclasses of starlike mappings along a unit direction on the unit polydisc. These results extend the classical distortion theorem of starlike mappings to higher dimensions. We then give an upper bound estimate for biholomorphic subclasses of starlike mappings along a unit direction in a complex Banach space.     

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.     

Characterization of isolated homogeneous hypersurface singularities in C4

Let V be a hypersurface with an isolated singularity at the origin in Cn+1. It is a natural question to ask when V is defined by weighted homogeneous polynomial or homogeneous polynomial up to biholomorphic change of coordinates. In 1971, a beautiful theorem of Saito gives a necessary and sufficient condition for V to be defined by a weighted homogeneous polynomial.For a two-dimensional isolated hypersurface singularity V, Xu and Yau found a coordinate free characterization for V to be defined by a homogeneous polynomial. Recently Lin and Yau gave necessary and sufficient conditions for a 3-dimensional isolated hypersurface singularity with geometric genus bi.er than zero to be defined by a homogeneous polynomial. The purpose of this paper is to prove that Lin-Yau's theorem remains true for singularities with geometric genus equal to zero.     

Characterization of isolated homogeneous hypersurface singularities in C~4 Dedicated to Professor Sheng GONG on the occasion of his 75th birthday

Let V be a hypersurface with an isolated singularity at the origin in Cn 1. It is a natural question to ask when V is defined by weighted homogeneous polynomial or homogeneous polynomial up to biholomorphic change of coordinates. In 1971, a beautiful theorem of Saito gives a necessary and sufficient condition for V to be defined by a weighted homogeneous polynomial. For a two-dimensional isolated hypersurface singularity V, Xu and Yau found a coordinate free characterization for V to be defined by a homogeneous polynomial. Recently Lin and Yau gave necessary and sufficient conditions for a 3-dimensional isolated hypersurface singularity with geometric genus bigger than zero to be defined by a homogeneous polynomial. The purpose of this paper is to prove that Lin-Yau's theorem remains true for singularities with geometric genus equal to zero.     

DISTORTION THEOREM FOR BIHOLOMORPHIC MAPPINGS IN TRANSITIVE DOMAINS (III)

The distortion theorem for biholomorphic convex mappings in bounded symmetric domains are considered. Especially the distortion theorem for biholomorphic convex mappings in classical domain of type IV and two exceptional domains are given.