Ancient Solutions to Curve Shortening with Finite Total Curvature

TL;DR

This paper constructs ancient solutions to the curve shortening flow in the plane with bounded total curvature by gluing Grim Reaper solutions along their asymptotes, expanding understanding of long-time behavior.

Contribution

It introduces a novel method of constructing ancient solutions with finite total curvature through gluing Grim Reapers, providing new examples in geometric flow theory.

Findings

Constructed explicit ancient solutions with finite total curvature.

Demonstrated a new gluing technique for Grim Reapers.

Extended the class of known solutions to curve shortening flow.

Abstract

We construct ancient solutions to Curve Shortening in the plane whose total curvature is uniformly bounded by gluing together an arbitrary chain of given Grim Reapers along their common asymptotes.