https://neckpinch.com/authors/kai-wei-zhao/New arXiv papers by Kai-Wei Zhao, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2603.09112arxiv:2603.09112Tue, 10 Mar 2026 00:00:00 GMTKyeongsu Choi, Dong-Hwi Seo, Wei-Bo Su, Kai-Wei Zhao — We prove that any ancient smooth embedded finite-entropy curve shortening flow is one of the following: a static line, a shrinking circle, a paper clip, a translating grim reaper, or a graphical ancient trombone. An ancient trombone is an immersed ancient flow, either compact or non-compact, obtained by gluing together m translating grim reaper curves. For each m, there exists a (2m-1)-parameter family of graphical ancient trombones, up to rigid motions and time shifts as constructed by Angenent-You. In particular, our result implies that any compact ancient smooth embedded finite-entropy flow is convex. Moreover, any non-compact ancient smooth embedded finite-entropy flow is either a static line or a complete graph over a fixed open interval.Curve shorteningAncient solutionsSelf-shrinkers & translatorsBoundaries & non-compact https://arxiv.org/abs/2410.01924arxiv:2410.01924Wed, 02 Oct 2024 00:00:00 GMTWei-Bo Su, Kai-Wei Zhao — Bounds of total curvature and entropy are two common conditions placed on mean curvature flows. We show that these two hypotheses are equivalent for the class of ancient complete embedded smooth planar curve shortening flows, which are one-dimensional mean curvature flows. As an application, we give a short proof of the uniqueness of tangent flow at infinity of an ancient smooth complete non-compact curve shortening flow with finite entropy embedded in ℝ².Mean curvature flowCurve shorteningAncient solutionsSingularitiesStability & uniquenessBoundaries & non-compact https://arxiv.org/abs/2405.10664arxiv:2405.10664Fri, 17 May 2024 00:00:00 GMTKyeongsu Choi, Dong-Hwi Seo, Wei-Bo Su, Kai-Wei Zhao — In this paper, we 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 its rescaled flows backwardly converge to a line with multiplity m≥ 3 exponentially fast in any compact region, unless the flow is a shrinking circle, a static line, a paper clip, or a translating grim reaper. In addition, we figure out the exact numbers of tips, vertices, and inflection points of the curves at negative enough time. Moreover, the exponential growth rate of graphical radius and the convergence of vertex regions to grim reaper curves will be shown.Curve shorteningAncient solutionsSelf-shrinkers & translatorsSingularitiesStability & uniquenessLong-time existence & convergence