This grant will be used to research problems in Differential Geometry. In particular, we will investigate the uniformization theory for positively curved complete Kaehler manifolds, the geometry and topology of nonpositively curved Riemannian manifolds with degenerate Ricci tensor, and certain Chern number inequalities for nonpositively curved compact Kaehler manifolds. The intellectual merit of this project derives in part from the central importance of these problems to the general theory of Differential Geometry for real and complex manifolds. We have previously worked on these topics and obtained some results, and we believe our specific approaches of this project offer the promise of important progress.
We believe that the project will have broad impact to the basic theory of Differential Geometry for Kaehler and Riemannian manifolds. Our approach here is to study some specific but very representative problems. Any progress made will advance the understanding in the area of Differential Geometry and related areas such as Topology, Geometric Analysis, Several Complex Variables and Algebraic Geometry. These areas are some of the major branches of the contemporary mathematics.
