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

TL;DR

This paper proves that proper holomorphic mappings between certain unbounded Fock-Bargmann-Hartogs domains are highly rigid, showing that such mappings are essentially automorphisms when the domain dimension satisfies specific conditions.

Contribution

It establishes a rigidity theorem for proper holomorphic mappings between Fock-Bargmann-Hartogs domains, extending understanding of their automorphism groups.

Findings

Proper holomorphic self-maps are automorphisms for $m \,\geq\, 2$.

Rigidity results hold for equidimensional Fock-Bargmann-Hartogs domains.

Automorphism groups are characterized for these domains.

Abstract

The Fock-Bargmann-Hartogs domain $D_{n,m}(\mu)$ ($\mu>0$) in $\mathbf{C}^{n+m}$ is defined by the inequality $\|w\|^2<e^{-\mu\|z\|^2},$ 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 $m\geq 2$ must be an automorphism.