Convex Ancient Solutions to Anisotropic Curve Shortening Flow

TL;DR

This paper constructs unique translating and ancient solutions to anisotropic curve shortening flow, demonstrating their properties and uniqueness within specific geometric constraints.

Contribution

It introduces the first known ancient compact solutions to anisotropic curve shortening flow and establishes their uniqueness alongside translating solutions.

Findings

Constructed a translating solution for anisotropic curve shortening flow.

Proved the uniqueness of the ancient compact solution within a given slab.

Demonstrated the solutions' properties and their role in the flow's geometric constraints.

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 no smaller.