Hua domain, named after Chinese mathematician Loo-Keng Hua, is defined as a domain in $${\mathbb {C}}^{n}$$
 fibered over an irreducible bounded symmetric domain $$\Omega \subset {\mathbb {C}}^{d}$$
 with the fiber over $$z\in \Omega $$
 being a $$(n-d)$$
 -dimensional generalized complex ellipsoid $$\Sigma (z)$$
 . In general, a Hua domain is a nonhomogeneous bounded pseudoconvex domain without smooth boundary. The purpose of this paper is twofold. Firstly, we obtain what seems to be the first rigidity results on proper holomorphic mappings between two equidimensional Hua domains. Secondly, we determine the explicit form of the biholomorphisms between two equidimensional Hua domains. As a special conclusion of this paper, we completely describe the group of holomorphic automorphisms of the Hua domain.

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Rigidity of proper holomorphic mappings between equidimensional Hua domains

In this paper, authors investigate the order of growth and the hyper order of solutions of a class of the higher order linear differential equation, and improve results of M. Ozawa^[6], G. Gundersen^[7] and J.K. Langley^[8], Li Chun-hong^[11].

On the growth of solutions of a class of higher order differential equations

We consider recent developments in the study of automorphism groups of domains in complex space. Particular attention is paid to results with a basis in geometry. Keywords: Domains, automorphism group, automorphism, invariant metrics, complex differential geometry Quaestiones Mathematicae 36(2013), 225–251

The automorphism groups of domains in complex space: a survey

Hua domain, named after Chinese mathematician Loo-Keng Hua, is defined as a domain in $\mathbb{C}^{n}$ fibered over an irreducible bounded symmetric domain $\Omega\subset \mathbb{C}^{d}\;(d<n)$ with the fiber over $z\in \Omega$ being a $(n-d)$-dimensional generalized complex ellipsoid $\Sigma(z)$. In general, a Hua domain is a nonhomogeneous domain without smooth boundary. The purpose of this paper is twofold. Firstly, we obtain what seems to be the first rigidity results on proper holomorphic mappings between two equidimensional Hua domains. Secondly, we determine the explicit form of the biholomorphisms between two equidimensional Hua domains. As a special conclusion of this paper, we completely describe the group of holomorphic automorphisms of the Hua domain.

The definition of classical holomorphic function spaces such as the Hardy space or the Dirichlet space on the Hartogs triangle is not canonical. In this paper we introduce a natural family of holomorphic function spaces on the Hartogs triangle which includes some weighted Bergman spaces, a candidate Hardy space and a candidate Dirichlet space. For the weighted Bergman spaces and the Hardy space we study the $L^p$ mapping properties of Bergman and Szegő projection respectively, whereas for the Dirichlet space we prove it is isometric to the Dirichlet space on the bidisc.

Holomorphic function spaces on the Hartogs triangle.

In this paper, we completely determine the structure of the holomorphic automorphism group of a generalized complex ellipsoid. This is a natural generalization of a result due to Landucci. Also, this gives an affirmative answer to an open problem posed by Jarnicki and Pflug.

On the holomorphic automorphism group of a generalized complex ellipsoid

In the study of the holomorphic automorphism group Aut( ) of a complex manifold , it seems to be natural to direct our attention not only t o the abstract group structure of Aut( ) but also to its topological group structu re equipped with the compact-open topology. In fact, a well-known theorem of H. C artan says that the topological group of the holomorphic automorphisms of a bounded omain in C has the structure of a Lie group, and this result enables us to make va rious kinds of detailed studies of bounded domains in C . On the other hand, in contrast to the case of bounded domains, the holomorphic automorphism group Aut (C × (C) ) of the unbounded domainC ×(C∗) is terribly big when +≥ 2, and cannot have the structure of a Lie group. But, by looking at topological subgroups of Au t(C × (C) ) with Lie group structures, we can find a lead to apply the Lie group theo ry t the investigation of the problems related to the structure of Aut( C × (C) ). In the present paper, we try to approach from this standpoint to the fundamental pr oblem of what complex manifold has the holomorphic automorphism group isomorphi c to Aut(C × (C) ) as topological groups. Namely, we prove the following result w i h the aid of the theory of Reinhardt domains developed in Shimizu [8], [9] (cf. Kruz hilin [6]).

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A group-theoretic characterization of the space obtained by omitting the coordinate hyperplanes from the complex Euclidean space

In this paper, we prove that the holomorphic automorphism groups of the spaces C k × ( C * ) n - k and ( C k - { 0 } ) × ( C * ) n - k are not isomorphic as topological groups. By making use of this fact, we establish the following characterization of the space C k × ( C * ) n - k : Let M be a connected complex manifold of dimension n that is holomorphically separable and admits a smooth envelope of holomorphy. Assume that the holomorphic automorphism group of M is isomorphic to the holomorphic automorphism group of C k × ( C * ) n - k as topological groups. Then M itself is biholomorphically equivalent to C k × ( C * ) n - k . This was first proved by us in [5] under the stronger assumption that M is a Stein manifold.

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A group-theoretic characterization of the space obtained by omitting the coordinate hyperplanes from the complex Euclidean space, II

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An intrinsic characterization of the unit polydisc

Abstract Let M be a connected Stein manifold of dimension n. We show that if the holomorphic automorphism group of M is isomorphic to the holomorphic automorphism group of B k × ℂ n–k as topological group, then M itself is biholomorphically equivalent to B k × ℂ n–k , where B k denotes the open unit ball in ℂ k .

Akio Kodama

Papers

On the holomorphic automorphism group of a generalized complex ellipsoid

A group-theoretic characterization of the space obtained by omitting the coordinate hyperplanes from the complex Euclidean space

A group-theoretic characterization of the space obtained by omitting the coordinate hyperplanes from the complex Euclidean space, II

An intrinsic characterization of the unit polydisc

A group-theoretic characterization of the direct product of a ball and a Euclidean space