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.

11 pages

Cited by in corpus (4)

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• Rigidity of proper holomorphic mappings between generalized Fock-Bargmann-Hartogs domains