Shrinkers of the area-preserving curve-shortening flow: Existence and saddle-point property

TL;DR

This paper studies homothetic solutions of the area-preserving curve-shortening flow, proving the existence of non-circular shrinkers, providing a classification scheme, and establishing a saddle-point property similar to classical results.

Contribution

It introduces a classification scheme for non-circular shrinkers of the area-preserving curve-shortening flow and proves their saddle-point property.

Findings

Existence of non-circular shrinkers for APCSF.

A partial classification scheme for these shrinkers.

A saddle-point property analogous to Abresch-Langer curves.

Abstract

We consider homothetic evolutions of the area-preserving curve-shortening flow (APCSF), that is, classical curve shortening flow with an additional non-local forcing term. By using known results on $\lambda$-curves, we prove the existence of non-circular shrinkers for this flow. In our first main result, we present a partial classification scheme, similar to the well-known Abresch-Langer classification for shrinkers of curve-shortening flow. Finally, we also deduce a saddle-point property for all non-circular (APCSF)-shrinkers analogous to the known saddle-point property of Abresch-Langer curves.