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

Abstract: We classify closed convex ancient alpha-curve shortening flows for sub-affine-critical powers alpha <= 1/3. In addition, we show that closed convex smooth finite entropy ancient alpha-curve shortening flows with alpha > 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 = 1/k(2)-1 with 3 <= k is an element of 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. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

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