Rigidity of the Holomorphic Automorphism of the Generalized Fock-Bargmann-Hartogs Domains

We study a class of typical Hartogs domains which is called a generalized Fock-Bargmann-Hartogs domain Dn,mp(μ). The generalized Fock-Bargmann-Hartogs domain is defined by inequality eμ‖z‖2∑j=1m|ωj|2p<1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${e^{\mu {{\left\| z \right\|}^2}}}\sum\limits_{j = 1}^m {{{\left| {{\omega _j}} \right|}^{2p}} < 1} $$\end{document}, 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 Dn,m/p(μ) becomes a holomorphic automorphism if and only if it keeps the function ∑j=1m|ωj|2peμ‖z‖2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum\limits_{j = 1}^m {{{\left| {{\omega _j}} \right|}^{2p}}{e^{\mu {{\left\| z \right\|}^2}}}} $$\end{document} invariant.