Nonconvex ancient solutions to Curve Shortening Flow

TL;DR

This paper constructs a new class of ancient solutions to planar curve shortening flow that are compact, embedded, and resemble a rotating Yin-Yang soliton truncated and closed by Grim Reaper segments, revealing complex long-term behaviors.

Contribution

It introduces a novel ancient solution to curve shortening flow combining features of Yin-Yang and Grim Reaper solitons, expanding understanding of possible solution behaviors.

Findings

Constructed a compact, embedded ancient solution.

Described the solution's approximation by Yin-Yang and Grim Reaper solitons.

Analyzed the solution's asymptotic behavior as time approaches negative infinity.

Abstract

We construct an ancient solution to planar curve shortening. The solution is at all times compact and embedded. For $t\ll0$ it is approximated by the rotating Yin-Yang soliton, truncated at a finite angle $\alpha(t) = -t$, and closed off by a small copy of the Grim Reaper translating soliton.