Mean curvature flow and Ricci flow

The proposed research is at the intersection of differential geometry, partial differential equations, calculus of variations, stochastic analysis and general relativity. Specifically, the main focus is on two geometric versions of the heat equation: the evolution of surfaces by their mean curvature, and the evolution of curved spaces by Hamilton’s Ricci flow. While many foundational results have been obtained on both flows, a central problem is that singularities will form in most relevant situations. The main goal of the proposed research is to improve our understanding of the formation of singularities under mean curvature flow and Ricci flow, and to develop methods to continue the flow beyond the first singular time.

A long term goal of my research on mean curvature flow is to construct solutions with surgery for general mean convex hypersurfaces, widely generalizing the estimates and the methodology that I developed in my prior work with Bruce Kleiner. I will investigate various topological applications of mean curvature flow with surgery, notably higher-dimensional Smale type conjectures about the topology of the moduli-space of embedded spheres.

A long term goal of my research on Ricci flow (mostly joint with Aaron Naber) is to develop a theory of generalized solutions that enable us to continue the flow through singularities. We plan to develop the theory of these weak solutions. I'll also investigate several applications, in particular several geometric-analytic conjectures that have been left open after Perelman's solution of the Poincare conjecture.