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

Guo, Ting ; Feng, Zhiming ; Bi, Enchao

Czechoslovak Mathematical Journal, Tome 71 (2021) no. 2, pp. 373-386.

Résumé

We study a class of typical Hartogs domains which is called a generalized Fock-Bargmann-Hartogs domain $D_{n,m}^{p}(\mu )$. The generalized Fock-Bargmann-Hartogs domain is defined by inequality ${\rm e}^{\mu \|z\|^{2}}\sum _{j=1}^{m}|\omega _{j}|^{2p}1$, where $(z,\omega )\in \mathbb {C}^n\times \mathbb {C}^m$. 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 $D_{n,m}^{p}(\mu )$ becomes a holomorphic automorphism if and only if it keeps the function $\sum _{j=1}^{m}|\omega _{j}|^{2p}{\rm e}^{\mu \|z\|^{2}}$ invariant.

Détail

DOI : 10.21136/CMJ.2020.0364-19

Classification : 32H35
Keywords: generalized Fock-Bargmann-Hartogs domain; holomorphic automorphism group