Rigidity of proper holomorphic mappings between equidimensional Hua domains

TL;DR

This paper establishes rigidity results for proper holomorphic mappings between equidimensional Hua domains and explicitly characterizes biholomorphisms and automorphism groups of these complex domains.

Contribution

It provides the first rigidity results for proper holomorphic maps between equidimensional Hua domains and determines their biholomorphisms explicitly.

Findings

Rigidity results for proper holomorphic mappings established.

Explicit form of biholomorphisms between Hua domains determined.

Holomorphic automorphism group of Hua domains fully described.

Abstract

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.