Two theorems on the Fock-Bargmann-Hartogs domains

Abstract

In this paper, we prove two mutually independent theorems on the family of Fock-Bargmann-Hartogs domains. Let $D_1$ and $D_2$ be two Fock-Bargmann-Hartogs domains in $\mathbb{C}^{N_1}$ and $\mathbb{C}^{N_2}$, respectively. In Theorem 1, we obtain a complete description of an arbitrarily given proper holomorphic mapping between $D_1$ and $D_2$ in the case where $N_1 = N_2$. Also, we shall give a geometric interpretation of Theorem 1. And, in Theorem 2, we determine the structure of $\text{Aut}(D_1\times D_2)$ using the data of $\text{Aut}(D_1)$ and $\text{Aut}(D_2)$ for arbitrary $N_1$ and $N_2$.