Complex Differential Geometry: Nonpositively Curved and Nonnegatively Curved Manifolds

这笔拨款将用于研究微分几何中的问题。特别地，我们将研究正弯曲完备Kaehler流形的均匀化理论，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流形的某些陈数不等式。这个项目的学术价值部分来自于这些问题对于实数和复数流形的微分几何一般理论的核心重要性。我们以前已经在这些课题上做过工作并取得了一些结果，我们相信这个项目的具体方法将提供重要的进展。我们相信这个项目将对Kaehler和Riemannian流形的微分几何基本理论产生广泛的影响。我们在这里的方法是研究一些具体但非常有代表性的问题。所取得的任何进展都将促进对微分几何及其相关领域如拓扑学、几何分析、多复变量和代数几何的理解。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.