某些Ricci曲率非负完备非紧流形的几何结构

该计划研究某些Ricci曲率非负的完备非紧流形的几何与拓扑。部分是前一计划（No.11571228）的继续：我们将继续研究具非负Ricci曲率且二次渐进非负截面曲率完备流形的几何与拓扑。在前面研究中，我们构造了五维及五维以上的相应流形的例子，它们有无限拓扑型，因而就五维及五维以上的情形否定了下述猜测：具非负Ricci曲率且二次渐进非负截面曲率的完备流形有有限拓扑型。因此目前计划的部分是集中研究四维情形的猜测。一个与此密切相关的问题是4-维Ricci-平坦完备非紧流形的结构，特别当流形是凯勒的时候。一个十分有趣的问题是：在什么样的条件下，一个复2维完备非紧Ricci-平坦的凯勒流形是一个紧复曲面中的一个除子的补；该计划也将致力于这些问题的探索。

This project is concerning the study of geometry and topology of some complete noncompact manifolds with nonnegative Ricci curvature. Its part is a continuation of the previous project (No.11571228): We'll go on study of geometry and topology of complete manifolds with non-negative Ricci curvature and quadratically asymptotically non-negative sectional curvature. In the previous study, we constructed the examples of such manifolds of dimension>4, but of infinite topological type, and hence disprove the following conjecture in the case of dimension>4: Any complete manifolds with nonnegative Ricci curvature and quardratically asymptotically nonnegative sectional curvature is of finite topological type. Therefore, a part of the present project will focus on the case of 4 dimension. A closely related problem to this is the structure of 4-dim complete noncompact Ricci-flat manifolds, in particular in the Kaehlerian case. A very interesting problem is under what condition a complex 2-dim complete noncompact Ricci-flat Kaehler manifold is the complement of a divisor in a compact complex surface; this project is also devoted to the exploration of these problems.