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

The Fock–Bargmann–Hartogs domain D n , m ( μ ) ( μ > 0 ) in C n + m is defined by the inequality ‖ w ‖ 2 < e − μ ‖ z ‖ 2 , where ( z , w ) ∈ C n × C m , which is an unbounded non-hyperbolic domain in C n + m . Recently, Yamamori gave an explicit formula for the Bergman kernel of the Fock–Bargmann–Hartogs domains in terms of the polylogarithm functions and Kim–Ninh–Yamamori determined the automorphism group of the domain D n , m ( μ ) . In this article, we obtain rigidity results on proper holomorphic mappings between two equidimensional Fock–Bargmann–Hartogs domains. Our rigidity result implies that any proper holomorphic self-mapping on the Fock–Bargmann–Hartogs domain D n , m ( μ ) with m ≥ 2 must be an automorphism.