Work

Uniqueness of Tangent Flows at Infinity for Finite-Entropy Shortening Curves

Year: 2025

Type: article

Abstract: In this paper, we prove that an ancient smooth curve-shortening flow with finite entropy embedded in $\mathbb{R}^{2}$ has a unique tangent flow at infinity. To this end, we show that its rescaled flows backwardly converge to a line with multiplicity 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.

Source: Geometric and Functional Analysis

DOI: 10.1007/s00039-025-00705-1

Authors: Kyeongsu Choi, Dong-Hwi Seo, Wei-Bo Su, Kai-Wei Zhao

Institutions: Korea Institute for Advanced Study, Seoul National University, National Center for Theoretical Sciences, National Center for Theoretical Sciences, Mathematics Division, University of Notre Dame

Language: English

Topic: Geometric Analysis and Curvature Flows

Subfield: Applied Mathematics

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $2,390