Spaces with Curvature Bounded from Below

University Of California-Santa Barbara, Santa Barbara CA

Investigators

Abstract

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.