The Curve Shortening Flow

Abstract

This is an expository paper describing the recent progress in the study of the curve shortening equation

$${X_{{t\,}}} = \,kN $$

((0.1))

Here X is an immersed curve in ℝ², k the geodesic curvature and N the unit normal vector. We review the work of Gage on isoperimetric inequalities, the work of Gage and Hamilton on the associated heat equation and the work of Epstein and Weinstein on the stable manifold theorem for immersed curves. Finally we include a new proof of the Bonnesen inequality and a proof that highly symmetric immersed curves flow under (0.1) to points.