On bounds of entropy and total curvature for ancient curve shortening flows

TL;DR

This paper proves the equivalence of bounds on total curvature and entropy for ancient planar curve shortening flows and uses this to establish the uniqueness of tangent flows at infinity.

Contribution

It demonstrates the equivalence of curvature and entropy bounds for ancient curve flows and provides a new proof of tangent flow uniqueness.

Findings

Total curvature and entropy bounds are equivalent for ancient planar curve flows.

Established the uniqueness of tangent flows at infinity for certain ancient flows.

Simplified proof of tangent flow uniqueness using these bounds.

Abstract

Bounds of total curvature and entropy are two common conditions placed on mean curvature flows. We show that these two hypotheses are equivalent for the class of ancient complete embedded smooth planar curve shortening flows, which are one-dimensional mean curvature flows. As an application, we give a short proof of the uniqueness of tangent flow at infinity of an ancient smooth complete non-compact curve shortening flow with finite entropy embedded in $\mathbb{R}^2$.