Rigidity of proper holomorphic mappings between certain unbounded non-hyperbolic domains

Abstract

The Fock-Bargmann-Hartogs domain $D_{n,m}(\mu )$ ($\mu >0$) in ${\mathbf{C}}^{n+m}$ is defined by the inequality $\Vert w{\Vert }^2<e^{-\mu \Vert z{\Vert }^2},$ where $(z,w)\in {\mathbf{C}}^n\times {\mathbf{C}}^m$, which is an unbounded non-hyperbolic domain in ${\mathbf{C}}^{n+m}$. Recently, Yamamori gave an explicit formula for the Bergman kernel of the Fock-Bargmann-Hartogs domains in terms of the polylogarithm functions and Kim-Ninh-Yamamori determined the automorphism group of the domain $D_{n,m}(\mu )$. In this article, we obtain rigidity results on proper holomorphic mappings between two equidimensional Fock-Bargmann-Hartogs domains. Our rigidity result implies that any proper holomorphic self-mapping on the Fock-Bargmann-Hartogs domain $D_{n,m}(\mu )$ with $m\ge 2$ must be an automorphism.

arXiv comment11 pages

References in corpus (1)

• The automorphism group of a certain unbounded non-hyperbolic domain

Cited by in corpus (7)

• Rigidity of proper holomorphic mappings between equidimensional Hua domains
• Proper holomorphic mappings between generalized Hartogs triangles
• Rigidity of Proper Holomorphic Self-mappings of the Pentablock
• Classification of proper holomorphic mappings between certain unbounded non-hyperbolic domains
• On the linearity of origin-preserving automorphisms of quasi-circular domains in ${\mathbb{C}}^n$
• Rigidity of proper holomorphic mappings between generalized Fock-Bargmann-Hartogs domains
• Balanced metrics on the Fock-Bargmann-Hartogs domains