Convex ancient solutions to anisotropic curve shortening flow

Abstract

We construct a translating solution to anisotropic curve shortening flow and show that for a given anisotropic factor $g:S^1\to {\mathbb{R}}_{+}$, and a given direction and speed, this translator is unique. We then construct an ancient compact solution to anisotropic curve shortening flow, and show that this solution, along with the appropriate translating solution, are the unique solutions to anisotropic curve shortening flow that lie in a slab of a given width, and in no smaller slab.