Classifying convex compact ancient solutions to the affine curve shortening 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.