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

Abstract

We classify closed convex α-curve shortening flows for sub-affine-critical powers α ≤ 13. In addition, we show that closed convex smooth finite entropy α-curve shortening flows with 13<α 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 α=1k2-1 with 3≤ k ∈ N, the round circle shrinker has non-trivial Jacobi fields, but the ancient flows do not evolve along the Jacobi fields.