Uniqueness of tangent flows at infinity for finite-entropy shortening curves

TL;DR

This paper proves the uniqueness of tangent flows at infinity for finite-entropy embedded curve shortening flows in the plane, revealing exponential convergence properties and detailed geometric features of the flow.

Contribution

It establishes the uniqueness of tangent flows at infinity for finite-entropy curves and characterizes their asymptotic behavior, including convergence rates and geometric features.

Findings

Rescaled flows converge exponentially to a line with multiplicity m≥3

Exact counts of tips, vertices, and inflection points at negative times

Growth rate of graphical radius and convergence to grim reaper curves

Abstract

In this paper, we prove that an ancient smooth curve shortening flow with finite-entropy embedded in $\mathbb{R}^2$ has a unique tangent flow at infinity. To this end, we show that its rescaled flows backwardly converge to a line with multiplity $m\geq 3$ exponentially fast in any compact region, unless the flow is a shrinking circle, a static line, a paper clip, or a translating grim reaper. In addition, we figure out the exact numbers of tips, vertices, and inflection points of the curves at negative enough time. Moreover, the exponential growth rate of graphical radius and the convergence of vertex regions to grim reaper curves will be shown.