Classifying convex compact ancient solutions to the affine curve shortening flow

TL;DR

This paper classifies all convex compact ancient solutions to the affine curve shortening flow as shrinking ellipses, using affine invariance, monotonicity, and rescaling techniques, with two different proofs provided.

Contribution

It provides a complete classification of convex compact ancient solutions to the affine curve shortening flow, establishing they are necessarily shrinking ellipses, and introduces two distinct proof methods.

Findings

All convex compact ancient solutions are shrinking ellipses.

Two different proofs are provided: one using level set representation, another employing Schauder's estimates.

The second proof also simplifies classification for higher-dimensional affine normal flow.

Abstract

In this paper we classify convex compact ancient solutions to the affine curve shortening flow: namely, any convex compact ancient solution to the affine curve shortening flow must be a shrinking ellipse. The method combines a rescaling argument inspired by \cite{Wang}, affine invariance of the equation and monotonicity of the affine isoperimetric ratio. We will give two proofs. The essential ideas are related, but the first one uses level set represenation of the evolution. The second proof employs Schauder's estimates, and it also provides a new simple proof for the corresponding classification result to the higher dimensional affine normal flow.