A Short Note on Contracting Self-Similar Solutions of the Curve Shortening Flow

TL;DR

This paper provides a concise geometric proof that the only closed embedded contracting self-similar solutions under the curve shortening flow are circles, reaffirming a classical result in geometric analysis.

Contribution

It offers a simple, intuitive proof of a well-known theorem using an idea originally introduced by Gage, enhancing understanding of self-similar solutions in curve shortening flow.

Findings

Circles are the only closed embedded contracting self-similar solutions.

The proof is short, geometric, and intuitive.

Reaffirms classical result with a new perspective.

Abstract

By the curve shortening flow, the only closed embedded contracting self-similar solutions are circles: we give a very short and intuitive geometric proof of this basic and classical result using an idea of Gage.