Lectures on mean curvature flow of surfaces

Abstract

Mean curvature flow is the most natural evolution equation in extrinsic geometry, and shares many features with Hamilton's Ricci flow from intrinsic geometry. In this lecture series, I will provide an introduction to the mean curvature flow of surfaces, with a focus on the analysis of singularities. We will see that the surfaces evolve uniquely through neck singularities and nonuniquely through conical singularities. Studying these questions, we will also learn many general concepts and methods, such as monotonicity formulas, epsilon-regularity, weak solutions, and blowup analysis that are of great importance in the analysis of a wide range of partial differential equations. These lecture notes are from summer schools at UT Austin and CRM Montreal, and also contain a detailed discussion of open problems and conjectures.

29 pages, includes discussion of open problems (v2: major updates in Section 5 and Section 6)

References in corpus (13)

• Evolution of curves and surfaces by mean curvature
• A Topological Property of Asymptotically Conical Self-Shrinkers of Small Entropy
• Currents and Flat Chains Associated to Varifolds, with an Application to Mean Curvature Flow
• A general regularity theory for weak mean curvature flow
• Compactness theory of the space of Super Ricci flows
• The moduli space of two-convex embedded spheres
• Uniqueness of asymptotically conical tangent flows
• Ricci flow and contractibility of spaces of metrics
• Asymptotic structure of self-shrinkers
• Local Entropy and Generic Multiplicity One Singularities of Mean Curvature Flow of Surfaces
• Lectures on mean curvature flow
• On Ilmanen's multiplicity-one conjecture for mean curvature flow with type-I mean curvature
• Bounded Diameter Under Mean Curvature Flow