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

Kyeongsu Choi; Liming Sun

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arXiv · 2012.01727

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

Kyeongsu Choi·Liming Sun

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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.

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Kyeongsu Choi, Liming Sun. 2020-12-03. Classification of ancient flows by sub-affine-critical powers of curvature in $\mathbb{R}^2$. https://arxiv.org/abs/2012.01727

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