RIGIDITY OF PROPER HOLOMORPHIC SELF-MAPPINGS OF SOME UNBOUNDED WEAKLY PSEUDOCONVEX HARTOGS DOMAIN

Abstract

This paper is concerned with the rigidity of proper holomorphic self-mappings of some unbounded weakly pseudoconvex domain, called a generalized Fock–Bargmann–Hartogs domain, which is defined as a Hartogs domain fibered over ${ℂ}^n$ with the fiber being a generalized complex ellipsoid. We develop a new technique to show that any proper holomorphic self-mapping of a generalized Fock–Bargmann–Hartogs domain must be an automorphism without the restriction of the dimension of each fiber. As a main contribution, we partly solve the rigidity problems for proper holomorphic self-mappings of generalized Fock–Bargmann–Hartogs domains.