A note on self-similar solutions of the curve shortening flow

TL;DR

This paper explores self-similar solutions of the curve shortening flow, providing an alternative ODE-based approach and confirming their planar nature, which aids understanding of singularity formation in geometric flows.

Contribution

It introduces a new method to analyze self-similar solutions via a simple ODE and offers an alternative proof of their planar property.

Findings

Self-shrinking and self-expanding solutions are characterized by a simple ODE.

Self-similar solutions are proven to lie in planes.

The approach relates to singularity formation in mean curvature flow.

Abstract

This article gives an alternative approach to the self-shrinking and self-expanding solutions of the curve shortening flow, which are related to singularity formation of the mean curvature flow. The motivation for the self-similar solutions arises from natural area preserving rescaling. Further we describe the self-similar solutions in terms of a simple ODE and give an alternative proof that they lie in planes.