Mean curvature flow and Ricci flow

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The project aims to study two geometric versions of the heat equation: the evolution of surfaces by their mean curvature, and the evolution of curved spaces by Ricci flow. Mean curvature flow models many physical processes which involve an evolving surface, or interface. It is the most efficient way to decrease the area of surfaces and to evolve them towards optimal ones. Correspondingly, Ricci flow deforms curved spaces towards optimal shapes. While many foundational results have been obtained on both flows, a central problem is that in most relevant situations singularities will form. The proposed research of the PI will provide ways to understand these singularities and to continue the flow through them. This will facilitate many new applications both within and outside mathematics.The proposed research is on mean curvature flow and Ricci flow, with a focus on the formation of singularities and techniques to continue the flow through singularities. In a joint work with Bruce Kleiner, the PI will give a new construction of mean curvature flow with surgery, based on their recent estimates for mean convex flows. The new construction is both more elementary and substantially shorter than prior ones. The PI will apply the mean curvature flow with surgery to topological problems, including a higher dimensional Smale conjecture. In a joint project with Aaron Naber, the PI will prove numerous estimates on path space for the Ricci flow. In fact, an evolving family of Riemannian manifolds satisfies these estimates if and only if it evolves by Ricci flow. Based on this, the PI and Naber will define a notion of weak solutions for the Ricci flow, and develop the theory of these weak solutions.