Czechoslovak Mathematical Journal, first online, pp. 1-14


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

Ting Guo, Zhiming Feng, Enchao Bi

Received August 12, 2019.   Published online April 17, 2020.

Abstract:  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 $\botsmash{\sum_{j=1}^m}|\omega_j|^{2p}{\rm e}^{\mu\|z\|^2}$ invariant.
Keywords:  generalized Fock-Bargmann-Hartogs domain; holomorphic automorphism group
Classification MSC:  32H35
DOI:  10.21136/CMJ.2020.0364-19

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Affiliations:   Ting Guo, School of Mathematics and Statistics, Qingdao University, 308 Ningxia Road, Qingdao, Shandong 266071, P. R. China, e-mail: 2018020209@qdu.edu.cn; Zhiming Feng, School of Mathematical and Information Sciences, Leshan Normal University, 778 Binhe Rd, Shizhong District, Leshan, Sichuan 614000, P. R. China, e-mail: fengzm2008@163.com; Enchao Bi (corresponding author), School of Mathematics and Statistics, Qingdao University, 308 Ningxia Road, Qingdao, Shandong 266071, P. R. China, e-mail: bienchao@whu.edu.cn


 
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