Classification of ancient flows by sub-affine-critical powers of curvature in R2

Abstract

We classify closed convex ancient α-curve shortening flows for sub-affine-critical powers $\alpha \le \frac{1}{3}$. In addition, we show that closed convex smooth finite entropy ancient α-curve shortening flows with $\alpha >\frac{1}{3}$ are shrinking circles. After rescaling, the ancient flows satisfying the above conditions converge exponentially fast to smooth closed convex shrinkers as the time goes to negative infinity. In particular, when $\alpha =\frac{1}{k^2-1}$ with $3\le k\in N$, the round circle shrinker has non-trivial Jacobi fields, but the ancient flows asymptotic to shrinking circles do not evolve along the Jacobi fields.