Characterization of the unit ball in {bf C}^n among complex manifolds of dimension n

We show that if the group of holomorphic automorphisms of a connected complex manifold $M$ of dimension $n$ is isomorphic as a topological group equipped with the compact-open topology to the automorphism group of the unit ball $B^n\subset\CC^n$, then $M$ is biholomorphically equivalent to either $B^n$ or $\CC\PP^n\setminus\bar{B^n}$.