Mean curvature flow and Ricci flow

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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. 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.