Manifolds with Lower Curvature Bounds and Their Limits

本文讨论了曲率界对具有不同截面曲率界和里奇曲率界的黎曼流形的局部几何性质和全局拓扑结构的影响及其Gromov-Hausdorff极限。主要研究者将研究满足自然几何假设的空间奇点的结构。了解这种奇点的结构通常可以提供关于这种空间的拓扑性质的大量信息。他们将研究Ricci曲率有界以下流形的gromov - hausdorff极限的结构；具有下截面曲率界的收敛拓扑具有低曲率界流形的基本群结构等周常数的最优界；单连通流形上非负曲率的阻碍。像流形这样的几何物体自然地出现在科学和工程中，作为构型空间，就像爱因斯坦的宇宙模型一样。里奇曲率是爱因斯坦广义相对论中的一个基本概念。因此，这些领域的基础研究不仅本身就很重要，而且应该对物理和工程产生影响。

This proposal is concerned with the effect of curvature bounds onthe local geometric properties and global topology of Riemannianmanifolds with various sectional and Ricci curvature bounds andof their Gromov-Hausdorff limits. The principal investigatorswill study the structure of the singularities that spacessatisfying some natural geometric assumptions candevelop. Understanding the structure of such singularities canoften give a lot of information about the topological propertiesof such spaces. They will study the structure of Gromov-Hausdorfflimits of manifolds with Ricci curvature bounded below; topologyof convergence with lower sectional curvature bound; thestructures of the fundamental groups for manifolds with lowercurvature bound; optimal bound of isoperimetric constant;obstructions to nonnegative curvature on simply-connectedmanifolds.Geometric objects such as manifolds appear naturally in scienceand engineering, as configuration spaces, as Einsten's model ofuniverse. Ricci curvature is a fundamental concept in Einstein'sgeneral relativity. Thus the fundamental research in these areashould not only be important in its own right but also shouldhave implications in physics and engineering.