An intrinsic characterization of the direct product of balls

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 B of balls in C n , then M itself is biholomorphically equivalent to B .