Rigidity of the holomorphic automorphism of the generalized Fock-Bargmann-Hartogs domains

Abstract

We study a class of typical Hartogs domains which is called a generalized Fock-Bargmann-Hartogs domain D (μ). The generalized Fock-Bargmann-Hartogs domain is defined by inequality $${e^{\mu {{\left\| z \right\|}^2}}}\sum\limits_{j = 1}^m {{{\left| {{\omega _j}} \right|}^{2p}} < 1} $$ , where (z, ω) ∈ ℂn × ℂm. In this paper, we will establish a rigidity of its holomorphic automorphism group. Our results imply that a holomorphic self-mapping of the generalized Fock-Bargmann-Hartogs domain D / (μ) becomes a holomorphic automorphism if and only if it keeps the function $$\sum\limits_{j = 1}^m {{{\left| {{\omega _j}} \right|}^{2p}}{e^{\mu {{\left\| z \right\|}^2}}}} $$ invariant.