Spaces with Curvature Bounded from Below

本研究项目研究具有Ricci曲率下界的流形。Ricci曲率是几何学家用来描述空间结构弯曲程度的方法之一，它是爱因斯坦场方程式和最优质量输运研究的重要组成部分。首席研究员将研究最基本的拓扑信息--基本群，它总结了空间中回路的行为。已知的关于Ricci曲率的点状界对空间的全局拓扑有重要影响，这些项目将寻求类似于较弱的关于Ricci曲率的整合界的结果.本文研究具有点态或积分Ricci曲率自下界的空间的几何和拓扑.许多几何问题导致积分曲率；例如，等谱问题、几何变分问题和极值度量，以及特征数的Chern-Weil公式。最近，这些条件也自然地出现在关于四维Kahler-Ricci流和Ricci流的工作中。主要研究者将研究具有积分Ricci下界的流形的第一本征值和基本群。PI还将研究满足黎曼曲率维度条件的度量度量空间。

This research project studies manifolds with Ricci curvature lower bound. Ricci curvature is one of the ways that geometers use to describe the degree of bending in the structure of a space, and it is an essential component of Einstein's field equation and the study of optimal mass transport. The principal investigator will study the most fundamental topological information -- the fundamental group, which summarizes the behavior of loops in a space. Pointwise bounds on Ricci curvature are known to have major consequences for the global topology of a space, and these projects will seek similar consequences for weaker, integrated bounds on Ricci curvature.This proposal studies the geometry and topology of spaces with pointwise or integral Ricci curvature bounded from below. Many geometric problems lead to integral curvatures; for example, isospectral problems, geometric variational problems and extremal metrics, and the Chern-Weil formula for characteristic numbers. Recently these conditions have also occurred naturally in work on Kahler-Ricci flow and Ricci flow in dimension 4. The principal investigator will study the first eigenvalue and fundamental group of manifolds with integral Ricci lower bound. The PI will also study metric measure spaces satisfying the Riemannian curvature dimension condition.