Curve shortening flow is, in some sense, the gradient flow of arc-length functional. It is the simplest geometric flow and is a special case of mean curvature flow. The classification problem of ancient solutions under some geometric conditions can be viewed as a parabolic analogue of Bernstein’s problem for minimal surfaces. The previous results technically rely on the assumption of convexity of the curves. In the joint project with Kyeongsu Choi, Donghwi Seo, and Weibo Su, we replace it by the boundedness of entropy, which a measure of geometric complexity defined by Colding and Minicozzi. In this talk, we will focus on the ancient solutions with entropy at most 2, which is the limiting model of “fingers” and “tails” of solutions with higher finite entropy.

 

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