Complex Differential Geometry: Nonpositively Curved and Nonnegatively Curved Manifolds

这笔赠款将用于研究微分几何问题。特别地，我们将研究正弯曲完备Kaehler流形的一致化理论，具有退化Ricci张量的非正弯曲黎曼流形的几何和拓扑，以及非正弯曲紧致Kaehler流形的某些Chern数不等式。 这个项目的智力价值部分来自于这些问题对真实的和复杂流形的微分几何一般理论的核心重要性。我们以前在这些课题上做过工作并取得了一些成果，我们相信我们的具体方法为这个项目提供了重要进展的希望。我们相信这个项目将对Kaehler和Riemann流形的微分几何基础理论产生广泛的影响。我们在这里的做法是研究一些具体的，但很有代表性的问题。所取得的任何进展都将促进对微分几何和相关领域如拓扑学、几何分析、多复变和代数几何的理解。这些领域是当代数学的一些主要分支。

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.