Plurisubharmonic Functions and the Structure of Complete Kähler Manifolds with Nonnegative Curvature

In this paper, we study global properties of continuous plurisubharmonic functions on complete noncompact Kähler manifolds with nonnegative bisectional curvature and their applications to the structure of such manifolds. We prove that continuous plurisubharmonic functions with reasonable growth rate on such manifolds can be approximated by smooth plurisubharmonic functions through the heat flow deformation. Optimal Liouville type theorem for the plurisubharmonic functions as well as a splitting theorem in terms of harmonic functions and holomorphic functions are established. The results are then applied to prove several structure theorems on complete noncompact Kähler manifolds with nonnegative bisectional or sectional curvature. 0. Introduction In this paper, we are interested in the class of complete noncompact Kähler manifolds with nonnegative holomorphic bisectional curvature. We shall first give a detailed study on the properties of heat flow with plurisubharmonic functions as initial data. Then we shall use the results to prove a Liouville theorem on plurisubharmonic functions and a splitting theorem related to harmonic and holomorphic functions. All these results will then be applied to obtain structure theorems on Kähler manifolds with nonnegative sectional or holomorphic bisectional curvature. The first author was partially supported by NSF grant DMS-0328624, USA. The second author was partially supported by Earmarked Grant of Hong Kong #CUHK4032/02P. Received 05/08/2003.