A comparison theorem for the isoperimetric profile under curve shortening flow

TL;DR

This paper establishes a comparison theorem for the isoperimetric profile under normalized curve shortening flow, enabling sharp curvature bounds and providing a new proof of Grayson's theorem without complex techniques.

Contribution

It introduces a novel comparison theorem for isoperimetric profiles that simplifies the proof of Grayson's theorem by avoiding blowup, Harnack estimates, and self-similar solution classification.

Findings

Derived sharp upper bounds on curvature for evolving curves.

Established lower bounds on curvature using the comparison theorem.

Provided a simplified proof of Grayson's theorem.

Abstract

We prove a comparison theorem for the isoperimetric profiles of simple closed curves evolving by the normalized curve shortening flow: If the isoperimetric profile of the region enclosed by the initial curve is greater than that of some `model' convex region with exactly four vertices and with reflection symmetry in both axes, then the inequality remains true for the isoperimetric profiles of the evolved regions. We apply this using the Angenent solution as the model region to deduce sharp time-dependent upper bounds on curvature for arbitrary embedded closed curves evolving by the normalized curve shortening flow. A slightly different comparison also gives lower bounds on curvature, and the result is a simple and direct proof of Grayson's theorem without use of any blowup or compactness arguments, Harnack estimates, or classification of self-similar solutions.