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Keywords:
generalized Fock-Bargmann-Hartogs domain; holomorphic automorphism group
generalized Fock-Bargmann-Hartogs domain; holomorphic automorphism group
Summary:
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.
References:
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