ncts

Curve shortening flow is, in the compact case, the gradient flow of arc-length functional. It is a one-dimensional case of mean curvature flow. The classification problem of ancient solutions under some geometric conditions is a parabolic version of geometric Liouville-type theorem. The previous results technically rely on the assumption of convexity of the curves. In the ongoing project joint with Kyeongsu Choi, Dong-Hwi Seo, and Wei-Bo Su, we replace the convexity condition by the boundedness of entropy, a measure of geometric complexity introduced by Colding and Minicozzi. In this talk, we show that an ancient smooth curve shortening flow with finite entropy embedded in R² has a unique tangent flow at infinity. Furthermore, we show that flow with entropy less than 3 must be a shrinking circle, a static line, a paper clip, or a translating grim reaper and show that if entropy m is greater than or equal to 3, then its rescaled flows backwardly converge to a line with multiplicity m exponentially fast in any compact region and m is an integer.