Complex Differential Geometry: Nonpositively Curved and Nonnegatively Curved Manifolds

这笔经费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全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，以及非正弯曲紧化Kaehler流形的某些Chern数不等式。这个项目的智力价值部分源于这些问题对实流形和复流形微分几何一般理论的核心重要性。我们以前曾就这些主题进行过研究，并取得了一些成果，我们相信，我们对这个项目的具体方法有望取得重要进展。我们相信该项目将对凯勒流形和黎曼流形的微分几何基础理论产生广泛的影响。我们的方法是研究一些具体但很有代表性的问题。任何进步都将促进对微分几何和相关领域的理解，如拓扑学、几何分析、几个复变量和代数几何。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.