Characterization of the Unit Ball in C Among Complex Manifolds of Dimension n

For a complex manifold M denote by Aut(M) the group of holomorphic automorphisms of M . Equipped with the compact-open topology, Aut(M) is a topological group. We are interested in characterizing complex manifolds by their automorphism groups. One manifold that has been enjoying much attention in this respect is the unit ball B ⊂ C for n ≥ 2. Starting with the famous theorems of Wong [W] and Rosay [R] many results characterizing B in terms of its automorphism group have been obtained. We mention proofs of Rosay’s theorem by means of invariant metrics [Kl], by means of scaling [P], by means of analyzing the structure of the ring of holomorphic functions [KK2], as well as extensions of the theorem to the case of unbounded domains [E], domains in complex manifolds [GKK] and domains (both bounded and unbounded) in infinitedimensional complex space [KK1], [BGK], [KM]. Rosay’s theorem implies, in particular, that a bounded homogeneous domain in C with C-smooth boundary is biholomorphically equivalent to B. A characterization result similar in spirit, but utilizing only the isotropy subgroup of a point in a complex manifold was obtained in [GK]. More information on results of this kind can be found in the survey [IKra1]. Among Kobayashi-hyperbolic manifolds, B can also be characterized as the manifold whose automorphism group has the largest dimension. Namely, if a connected complex manifold M of dimension n is hyperbolic, Aut(M)