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和Ricci曲率界的黎曼流形的局部几何性质和全局拓扑的影响，以及它们的Gromov-Hausdorff极限。主要研究人员将研究一些自然几何假设空间化可能产生的奇点的结构。了解这类奇点的结构通常可以提供关于这类空间的拓扑性质的许多信息。他们将研究Ricci曲率有界的流形的Gromov-Hausdorff极限的结构；具有较低截面曲率界限的收敛拓扑；具有较低曲率界限的流形的基本群的结构；等周常数的最优界；单连通流形上非负曲率的障碍。几何对象，如流形，自然地出现在科学和工程中，作为配置空间，就像Einsten的反模型一样。利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，而且还应该在物理和工程上产生影响。

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.