2019 Volume 71 Issue 4 Pages 559-580
Let $M$ be a connected Stein manifold of dimension $N$ and let $D$ be a Fock-Bargmann-Hartogs domain in $\mathbb{C}^N$. Let Aut$(M)$ and Aut$(D)$ denote the groups of all biholomorphic automorphisms of $M$ and $D$, respectively, equipped with the compact-open topology. Note that Aut$(M)$ cannot have the structure of a Lie group, in general; while it is known that Aut$(D)$ has the structure of a connected Lie group. In this paper, we show that if the identity component of Aut$(M)$ is isomorphic to 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 Aut$(D)$.
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