Lectures on mean curvature flow

TL;DR

This paper provides an introduction and unified overview of the theory of mean curvature flow, including recent advances in regularity, structure, and surgical techniques for hypersurfaces.

Contribution

It offers a streamlined, comprehensive treatment of White's and Huisken-Sinestrari's theories, integrating recent developments in mean curvature flow.

Findings

Unified framework for White's and Huisken-Sinestrari's theories

Regularity results for mean convex mean curvature flow

Flow with surgery for two-convex hypersurfaces

Abstract

A family of hypersurfaces evolves by mean curvature flow if the velocity at each point is given by the mean curvature vector. Mean curvature flow is the most natural evolution equation in extrinsic geometry, and has been extensively studied ever since the pioneering work of Brakke and Huisken. In the last 15 years, White developed a far-reaching regularity and structure theory for mean convex mean curvature flow, and Huisken-Sinestrari constructed a flow with surgery for two-convex hypersurfaces. In this course, I first give a general introduction to the mean curvature flow of hypersurfaces and then present joint work with Bruce Kleiner, where we give a streamlined and unified treatment of the theory of White and Huisken-Sinestrari. These notes are from summer schools at KIAS Seoul and SNS Pisa.