Rigidity of proper holomorphic mappings between generalized Fock–Bargmann–Hartogs domains

Enchao Bi and Zhenhan Tu

Vol. 297 (2018), No. 2, 277–297

DOI: 10.2140/pjm.2018.297.277

Abstract

A generalized Fock–Bargmann–Hartogs domain $D_n^{m,p}\left(\mu \right)$ is defined as a domain fibered over ${ℂ}^n$ with the fiber over $z\in {ℂ}^n$ being a generalized complex ellipsoid ${\Sigma }_z\left(m,p\right)$. In general, a generalized Fock–Bargmann–Hartogs domain is an unbounded nonhyperbolic domain without smooth boundary. The main contribution of this paper is as follows. By using the explicit formula of Bergman kernels of the generalized Fock–Bargmann–Hartogs domains, we obtain the rigidity results of proper holomorphic mappings between two equidimensional generalized Fock–Bargmann–Hartogs domains. We therefore exhibit an example of unbounded weakly pseudoconvex domains on which the rigidity results of proper holomorphic mappings can be built.

Keywords

automorphism groups, Bergman kernels, generalized Fock–Bargmann–Hartogs domains, proper holomorphic mappings

Mathematical Subject Classification 2010

Primary: 32A07, 32A25, 32H35, 32M05

Milestones

Received: 23 October 2016

Revised: 24 July 2017

Accepted: 23 March 2018

Published: 20 December 2018