Classification of ancient flows by sub-affine-critical powers of curvature in $\mathbb{R}^2$

TL;DR

This paper classifies convex curve shortening flows in the plane for certain powers of curvature, showing that for powers above a threshold, flows converge to circles, with detailed analysis of their stability and asymptotic behavior.

Contribution

It provides a complete classification of convex ancient flows for sub-affine-critical powers and analyzes the stability of circle shrinkers for super-critical powers.

Findings

Classified convex $rac{1}{3}$-power flows in $ extbf{R}^2$.

Proved convergence of flows to circles for $rac{1}{3}< ext{power}$.

Analyzed Jacobi fields and stability of circle shrinkers.

Abstract

We classify closed convex $\alpha$-curve shortening flows for sub-affine-critical powers $\alpha \leq \frac{1}{3}$. In addition, we show that closed convex smooth finite entropy $\alpha$-curve shortening flows with $\frac{1}{3}<\alpha$ is a shrinking circle. After normalization, the ancient flows satisfying the above conditions converge exponentially fast to smooth closed convex shrinkers at the backward infinity. In particular, when $\alpha=\frac{1}{k^2-1}$ with $3\leq k \in \mathbb{N}$, the round circle shrinker has non-trivial Jacobi fields, but the ancient flows do not evolve along the Jacobi fields.