Geometry, Analysis and Topology under Curvature Bounds

AbstractAward：DMS-9971833主要研究者：魏国芳这个建议是关于Ricci曲率和积分曲率界对黎曼流形的局部几何结构和整体拓扑的影响，主要研究（i）度量到正截面曲率的变形，（ii）等周常数的最佳界，（iii）具有点态或积分Ricci曲率下界的流形的基本群的结构;（iv）具有有界弱范数的流形的结构。几何对象，如流形，自然出现在科学和工程中，作为配置空间，作为爱因斯坦的宇宙模型。 Ricci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重要，而且在科学和工程方面也有意义。

AbstractAward: DMS-9971833Principal Investigator: Guofang WeiThis proposal is concerned with the effect of Ricci curvature andintegral curvature bounds on the local geometric structure andglobal topology of a Riemannian manifold.The principal investigator will study (i) deformation of metrics to positive sectional curvature, (ii) optimal bound of isoperimetric constant, (iii) the structures of the fundamental groups for manifolds with pointwise or integral lower Ricci curvature bound, and(iv) structures of manifolds with bounded weak norm. 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 science and engineering.