A class of nonpositively curved Kahler manifolds biholomorphic to the   unit ball in C^n
  Let (M,g) be a simply connected complete Kahler manifold with nonpositive sectional curvature. Assume that g has constant negative holomorphic sectional curvature outside a compact set. We prove that M is then biholomorphic to the unit ball in C^n, where dim M = n. 
Harish Seshadri and Kaushal Verma