Classification of compact ancient solutions to the curve shortening flow

This is the authors' abstract. We don't add key points for this paper.

We consider an embedded convex ancient solution Γₜ to the curve shortening flow in R². We prove that there are only two possibilities: the family Γₜ is either the family of contracting circles, which is a type I ancient solution, or the family of evolving Angenent ovals, which correspond to a type II ancient solution to the curve shortening flow. We also give a necessary and sufficient curvature condition for an embedded, closed ancient solution to the curve shortening flow to be convex.