Classification of proper holomorphic mappings between certain unbounded non-hyperbolic domains

The Fock-Bargmann-Hartogs domain $D_{n,m}(\mu)$ ($\mu>0$) in $\mathbb{C}^{n+m}$ is defined by the inequality $\|w\|^2<e^{-\mu\|z\|^2},$ where $(z,w)\in \mathbb{C}^n\times \mathbb{C}^m$, which is an unbounded non-hyperbolic domain in $\mathbb{C}^{n+m}$. Recently, Tu-Wang obtained the rigidity result that proper holomorphic self-mappings of $D_{n,m}(\mu)$ are automorphisms for $m\geq 2$, and found a counter-example to show that the rigidity result isn't true for $D_{n,1}(\mu)$. In this article, we obtain a classification of proper holomorphic mappings between $D_{n,1}(\mu)$ and $D_{N,1}(\mu)$ with $N<2n$.