Poisson Equation, Poincaré-lelong Equation and Curvature Decay on Complete Kähler Manifolds

In the first part of this work, the Poisson equation on complete noncompact manifolds with nonnegative Ricci curvature is studied. Sufficient and necessary conditions for the existence of solutions with certain growth rates are obtained. Sharp estimates on the solutions are also derived. In the second part, these results are applied to the study of curvature decay on complete Kähler manifolds. In particular, the Poincaré-Lelong equation on complete noncompact Kähler manifolds with nonnegative holomorphic bisectional curvature is studied. Several applications are then derived, which include the Steinness of the complete Kähler manifolds with nonnegative curvature and the flatness of a class of complete Kähler manifolds satisfying a curvature pinching condition. Liouville type results for plurisubharmonic functions are also obtained. 0. Introduction In this paper, we will discuss the Poisson equation on complete noncompact manifolds and derive some applications on Kähler manifolds. Let Mm be a complete noncompact Kähler manifold, where m ≥ 2 is the complex dimension. Assume M has nonnegative holomorphic bisectional curvature and has maximal volume growth such that the scalar curvature decays like r−2 where r is the distance from a fixed point. Then it was proved in [20] by Mok-Siu-Yau that one can solve the following Poincaré-Lelong equation (0.1) √−1∂∂u = ρ Received November 25, 2000, and, in revised form, June 28, 2001. The first author was partially supported by NSF grant DMS9970284, USA, the second author was partially supported by NSF of China, project 10001001, and the third author was partially supported by Earmarked Grant of Hong Kong #CUHK4217/99P.