Speaker: Kai-Wei Zhao
University of Notre Dame
Will give a Geometric Analysis Seminar entitled:
Uniqueness of Tangent Flows at Infinity for Finite-Entropy Shortening Curves
Abstract: Curve shortening flow is, in 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 reply on the assumption of convexity of the curves. In the ongoing project joint with Kyeongsu Choi, Donghwi Seo, and Weibo Su, we replace convexity condition by the boundedness of entropy, a measure of geometric complexity introduced by Colding and Minicozzi. In this talk, we will prove that an ancient smooth curve shortening flow with finite-entropy embedded in R2 has a unique tangent flow at infinity. To this end, 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 is greater than or equal to 3, its rescaled flows backwardly converge to a line with multiplicity m≥3 exponentially fast in any compact region.
Date: 09-12-2024
Time: 11:00 am
Location: 125 Hayes-Healy