Characterization of C by its Automorphism Group

Let M be a connected complex manifold of dimension n and let Aut(M) denote the group of holomorphic automorphisms of M . The group Aut(M) is a topological group equipped with the natural compact-open topology. We are interested in the problem of characterizing M by Aut(M). This problem becomes particularly intriguing when Aut(M) is infinite-dimensional. Let, for example, M = C and suppose that M ′ is such that Aut(M ) is isomorphic as a topological group to Aut(C); is it then true that M ′ is biholomorphically equivalent to C? In [IK] we gave a positive answer to the above question, and the proof there followed from a general classification of all connected n-dimensional complex manifolds that admit effective actions of the unitary group Un by holomorphic transformations. In this paper we give a simpler proof in the case of Stein manifolds. This proof does not require considering the whole group Un, but relies only on linearization of the induced action of the torus T ⊂ Un [BBD]. In this paper we prove the following theorem.