Kähler manifolds with curvature bounded from above by a decreasing function

Abstract:

Let M be a simply connected complete Kähler manifold. If M has curvature bounded from above by a certain positive decreasing function, then it is a Stein manifold, diffeomorphic to a euclidean space. This fact is a generalization of the well-known propositions for complete manifolds of nonpositive curvature and is shown by the aid of a Rauch comparison theorem for conjugate points together with a comparison theorem of Siu and Yau with respect to the Hessian of distance functions.