Ricci Flows through Singularities and Ricci Flows with Bounded Scalar Curvature

University Of California-Berkeley, Berkeley CA

Investigators

Abstract

A Ricci flow is a geometric process that may be used to improve a given geometry towards a more homogeneous one. Ricci flows have become the subject of intensive study, as they have been used to prove various long-standing conjectures, such as the Poincare and Geometrization Conjectures in dimension 3. The general expectation is that a Ricci flow produces a geometry in the limit that is in some sense inherent to the topology, i.e. the loose makeup, of the underlying space. However, usually a Ricci flow develops complicated singularities in finite time. In dimension 3 these singularities can be removed manually by so called surgeries and the flow can be continued beyond them. Recently, a new class of "singular Ricci flows" was introduced in dimension 3. These flows flow "automatically through singularities at an infinitesimal scale", thereby eliminating the somewhat unnatural surgery process. The goal of this project is to understand these flows further and to use this understanding to study the topology of certain spaces of metrics and diffeomorphism groups. In addition, the PI will work on Ricci flows in higher dimensions, aimed at understanding their singularity formation, which may result in a similar surgery or singular flow construction. The research project is split into two projects. The first project is a continuation of the PI's work (in collaboration with Bruce Kleiner) on the uniqueness and continuity of singular Ricci flows through singularities. The general goal of this project is to understand the geometric, topological and analytic applications of this work. Among other things, the PI has a strategy to resolve the Generalized Smale Conjecture, which would extend a previous partial resolution by the PI and Kleiner. Further potential applications concern the topology and geometry of the space of positive scalar curvature metrics, as well as the study of generic Ricci flows in dimension 3. The second project is a continuation of the PI's work on the study of Ricci flows with bounded scalar curvature. The PI will investigate several conjectures that have been verified under the assumption of bounded scalar curvature. These conjectures are likely to remain true if this assumption is removed. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.