Lectures on curve shortening flow

TL;DR

This paper provides an accessible exposition of the curve shortening flow, illustrating key PDE concepts and presenting Huisken's proof that it transforms any closed embedded plane curve into a round point.

Contribution

It offers a comprehensive overview of the curve shortening flow and details Huisken's proof of Grayson's theorem, connecting geometric PDE techniques.

Findings

Curve shortening flow shrinks any closed embedded plane curve to a round point.

The paper illustrates maximum principle estimates, monotonicity formulas, and blowup analysis.

Huisken's proof of Grayson's theorem is systematically explained.

Abstract

The curve shortening flow is a geometric heat equation for curves and provides an accessible setting to illustrate many important concepts from nonlinear partial differential equations, including maximum principle estimates, monotonicity formulas, Harnack inequalities and blowup analysis. All these techniques will be combined to give an exposition of Huisken's proof of Grayson's beautiful theorem that the curve shortening flow shrinks any closed embedded curve in the plane to a round point.