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

## TL;DR

This paper establishes rigidity results for proper holomorphic mappings between generalized Fock-Bargmann-Hartogs domains by explicitly computing Bergman kernels, demonstrating the mappings' structural constraints in these complex, unbounded, and non-hyperbolic domains.

## Contribution

It provides the first explicit formula for Bergman kernels on these domains and proves rigidity of proper holomorphic mappings, extending understanding of complex mappings in unbounded non-hyperbolic domains.

## Key findings

- Rigidity results for proper holomorphic mappings between generalized Fock-Bargmann-Hartogs domains.
- Explicit formulas for Bergman kernels on these domains.
- Identification of unbounded weakly pseudoconvex domains with mapping rigidity.

## Abstract

A generalized Fock-Bargmann-Hartogs domain $D_n^{\mathbf{m},\mathbf{p}}$ is defined as a domain fibered over $\mathbb{C}^{n}$ with the fiber over $z\in \mathbb{C}^{n}$ being a generalized complex ellipsoid $\Sigma_z({\mathbf{m},\mathbf{p}})$. In general, a generalized Fock-Bargmann-Hartogs domain is an unbounded non-hyperbolic domains 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.