曲率几乎非负的紧致Kahler流形的几何与拓扑

This proposal is concerned about the geometry and topology of almost non-negatively curved compact Kahler manifolds, which certainly will enrich the knowledge of the geometry and topology of manifolds with curvature bounded from below. We intend to understand the structure of non-negatively curved(nef) manifolds more deeply. We will combine the methods of geometric flow and Gromov-Hausdorff theory to study the properties of manifolds with almost non-negative curvature. We are going to study the following problems:(1)Starting from the geometric analysis view, we will find out if there exists a Kähler metric with positive Ricci curvature for a simply connected compact Kähler manifold with almost non-negative holomorphic bisectional curvature ;(2)To study the structure of manifolds with almost non-negative orthogonal holomorphic bisectional curvature;(3)To study the structure of the limit space of a sequence of compact Kahler manifolds with almost non-negative curvature;(4)To study the geometry of compact Sasaki manifolds with almost nonnegative transverse bisectional curvature .

本项目针对曲率几乎非负紧致Kahler流形的性质进行研究,旨在丰富曲率有下界流形的几何与拓扑性质的理论内容,也希望更深入地理解曲率有下界流形的几何。我们将结合几何流, 特别是Ricci流与Gromov-Hausdorff理论的研究办法,深入研究曲率几乎非负(nef)紧致Kahler流形的性质。本项目拟就如下几个问题进行讨论:(1)研究几乎非负正交全纯双截曲率流形的几何结构;(2)从分析的角度出发,探讨具有几乎非负全纯双截曲率的紧致单连通Kähler流形上正Ricci曲率的Kähler度量的存在性; (3)研究一族曲率几乎非负紧致Kahler流形的极限空间的几何性质;（4）研究具有几乎非负横截全纯双截曲率的紧致Sasaki流形的微分结构.