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 $\mathrm{\parallel }w{\mathrm{\parallel }}^2<e^{-\mu \mathrm{\parallel }z{\mathrm{\parallel }}^2},$ 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 $m\ge 2$, 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 $N<2n$.