#### Vol. 297, No. 2, 2018

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

### Enchao Bi and Zhenhan Tu

Vol. 297 (2018), No. 2, 277–297
##### 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