Classification of ancient finite-entropy curve shortening flows

TL;DR

This paper classifies all ancient finite-entropy smooth embedded curve shortening flows, revealing they are limited to specific known types, and establishes convexity for compact flows and a simple structure for non-compact flows.

Contribution

It provides a complete classification of ancient finite-entropy curve shortening flows, including new constructions of ancient trombones and their parameter families.

Findings

All ancient finite-entropy flows are static lines, circles, paper clips, grim reapers, or ancient trombones.

Any compact ancient flow is convex.

Non-compact flows are either lines or graphs over an interval.

Abstract

We prove that any ancient smooth embedded finite-entropy curve shortening flow is one of the following: a static line, a shrinking circle, a paper clip, a translating grim reaper, or a graphical ancient trombone. An ancient trombone is an immersed ancient flow, either compact or non-compact, obtained by gluing together $m$ translating grim reaper curves. For each $m$, there exists a $(2m-1)$-parameter family of graphical ancient trombones, up to rigid motions and time shifts as constructed by Angenent-You. In particular, our result implies that any compact ancient smooth embedded finite-entropy flow is convex. Moreover, any non-compact ancient smooth embedded finite-entropy flow is either a static line or a complete graph over a fixed open interval.