Lectures on mean curvature flow of surfaces

TL;DR

This paper introduces the mean curvature flow of surfaces, focusing on singularity analysis, and discusses key concepts and open problems in the evolution of surfaces under this geometric flow.

Contribution

It provides an accessible introduction to mean curvature flow, emphasizing singularity analysis and the application of various analytical techniques.

Findings

Surfaces evolve uniquely through neck singularities.

Surfaces evolve nonuniquely through conical singularities.

The lecture notes include open problems and conjectures.

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.