Ricci Curvature, Ricci Flow and Foliations

University Of California-Berkeley, Berkeley CA

Investigators

Abstract

This proposal deals with curved spaces of higher dimension. An interesting question is how to measure the curvature of a nonsmooth space. In the case of sectional curvature, this was initiated in the 1930's by Alexandrov, who gave a good notion of what it means for a singular space to have nonnegative sectional curvature. Recent work by the proposer, joint with Cedric Villani, together with related work by Karl-Theodor Sturm, has given a good notion of what it means for a singular space to have nonnegative Ricci curvature. The proposed research will explore properties of spaces with nonnegative Ricci curvature or, more generally, with Ricci curvature bounded below. In a different direction, the Ricci curvature gives a way to smooth out the geometry of a space, by means of the Ricci flow. This flow was introduced by Richard Hamilton in the 1980's, who used it to characterize the topology of three-dimensional smooth spaces with nonnegative Ricci curvature. Recently, Perelman has proved the biggest conjectures in three-dimensional topology, namely the Poincare Conjecture and Thurston's Geometrization Conjecture, using Ricci flow. Despite Perelman's great achievements, there are many open questions concerning the Ricci flow in dimension three and in higher dimensions. The proposed research will address some of these questions. Another way that singular spaces arise is when a higher-dimensional space is foliated into lower dimensional spaces. The parametrizing space for such a foliation is almost always topologically singular. Part of the proposed research is to do analysis on such spaces, or more precisely to prove a transverse index theorem. There are various notions of curvature, which coincide in the traditional setting of two-dimensional surfaces in three-dimensional space. For a higher dimensional smooth space, not necessarily living in a flat space, these different notions are called the sectional curvature, the Ricci curvature and the scalar curvature. Each one is an averaging of the previous one, i.e. the Ricci curvature is an averaging of sectional curvature and the scalar curvature is an averaging of Ricci curvature. The Ricci curvature enters in physics through Einstein's equations of general relativity. Part of the work described above concerns ``optimal transport''. This is the study of the optimal way to transport mass in a curved space. It was initiated by Monge in the 1780's and has had a revival in recent decades, with application to partial differential equations and applied mathematics. We have shown that it also has application in differential geometry. Conversely, concepts from differential geometry have application to optimal transport.