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

Abstract

We classify closed convex $\alpha$-curve shortening flows for sub-affine-critical powers $\alpha \le \frac{1}{3}$. 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 $3\le k\in \mathbb{N}$, the round circle shrinker has non-trivial Jacobi fields, but the ancient flows do not evolve along the Jacobi fields.

References in corpus (5)

• Ancient gradient flows of elliptic functionals and Morse index
• Uniqueness of closed self-similar solutions to the Gauss curvature flow
• Unique Asymptotics of Compact Ancient Solutions to three-dimensional Ricci flow
• Mean curvature flow with generic initial data
• Ancient finite entropy flows by powers of curvature in ${\mathbb{R}}^2$

Cited by in corpus (1)

• Translating surfaces under flows by sub-affine-critical powers of Gauss curvature