Tohoku Mathematical Journal
2019
December
SECOND SERIES VOL. 71, NO. 4

Tohoku Math. J.
71 (2019), 559-580

Title A GROUP-THEORETIC CHARACTERIZATION OF THE FOCK-BARGMANN-HARTOGS DOMAINS

Author Akio Kodama

(Received October 27, 2017, revised January 26, 2018)
Abstract. Let $M$ be a connected Stein manifold of dimension $N$ and let $D$ be a Fock-Bargmann-Hartogs domain in $\mathbb{C}^N$. Let $\text{\rm Aut}(M)$ and $\text{\rm Aut}(D)$ denote the groups of all biholomorphic automorphisms of $M$ and $D$, respectively, equipped with the compact-open topology. Note that $\text{\rm Aut}(M)$ cannot have the structure of a Lie group, in general; while it is known that $\text{\rm Aut}(D)$ has the structure of a connected Lie group. In this paper, we show that if the identity component of $\text{\rm Aut}(M)$ is isomorphic to $\text{\rm Aut}(D)$ as topological groups, then $M$ is biholomorphically equivalent to $D$. As a consequence of this, we obtain a fundamental result on the topological group structure of $\text{\rm Aut}(D)$.

Mathematics Subject Classification. Primary 32A07; Secondary 32M05.
Key words and phrases. Fock-Bargmann-Hartogs domains, biholomorphic mappings, holomorphic automorphisms, stein manifolds.

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