A group-theoretic characterization of the Fock-Bargmann-Hartogs domains

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 $\mathrm{Aut}(M)$ and $\mathrm{Aut}(D)$ denote the groups of all biholomorphic automorphisms of $M$ and $D$, respectively, equipped with the compact-open topology. Note that $\mathrm{Aut}(M)$ cannot have the structure of a Lie group, in general; while it is known that $\mathrm{Aut}(D)$ has the structure of a connected Lie group. In this paper, we show that if the identity component of $\mathrm{Aut}(M)$ is isomorphic to $\mathrm{Aut}(D)$ 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 $\mathrm{Aut}(D)$.