An intrinsic characterization of the direct product of balls

Abstract

In this paper, we give a characterization of the direct product of balls by its holomorphic automorphism group. Using a result on the standardization of certain compact group actions on complex manifolds, we show that, for a connected Stein manifold $M$ of dimension $n$, if its holomorphic automorphism group contains a topological subgroup that is isomorphic to the holomorphic automorphism group of the direct product $\mathbf B$ of balls in $\mathbf C^n$, then $M$ itself is biholomorphically equivalent to $\mathbf B$.