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.
Categories: math.CV