Which shapes can appear in a curve shortening flow singularity?
type: Item
instance of: scholarly article
title: Which shapes can appear in a curve shortening flow singularity?
author: S. B. Angenent
author: Evan Patrick Davis
author: Ellie DeCleene
author: Paige Ellingson
author: Ziheng Feng
author: Q6203191
author: Aris Lemmenes
author: Alex Moon
author: Tyler Joseph Tommasi
author: Yamin Zhou
published in: Nonlinearity
publication date: 2024-11-05T00:00:00Z
review text: A family of closed curves is given by a differentiable map \(\gamma :[0,T)\times \mathbb{R}\rightarrow \mathbb{R}^{2}\), where \(u\rightarrow \gamma (t,u)\) parametrizes the curve at time \(t\in \lbrack 0,T)\), the curves satisfying \(\gamma (t,u+1)=\gamma (t,u)\) for all \(t,u\).\ The curves are immersed if \(\gamma _{u}(t,u)(=\frac{\partial \gamma }{\partial u})\neq 0\) for all \(t,u\). A family \(\gamma (t,\cdot )\) of curves evolves by curve shortening flow if it satisfies \[\gamma _{t}(t,u)^{\perp }=k(t,u),\] where \( \gamma _{t}^{\perp }\) is the component of \(\gamma _{t}\) that is perpendicular to \(\gamma _{u}\). The map \(\gamma :[0,T)\times \mathbb{R} \rightarrow \mathbb{R}^{2}\) is called a normal parametrization if \(\gamma _{t}\perp \gamma _{u}\) for all \((t,u)\). \N\NThe authors recall that for any initial curve \(\gamma ^{0}\) there is a unique solution to the curve shortening flow starting from \(\gamma ^{0}\). The maximal time interval \([0,T)\) during which the solution \(\gamma \) is defined depends on the initial curve, but it is always finite, the maximal curvature on the curve becoming infinite. \N\NThe authors recall the Gage-Hamilton-Grayson theorem for embedded curves [\textit{M. Gage} and \textit{R. S. Hamilton}, J. Differ. Geom. 23, 69--96 (1986; Zbl 0621.53001)], [\textit{M. A. Grayson}, J. Differ. Geom. 26, 285--314 (1987; Zbl 0667.53001)] and Oaks' theorem for immersed curves [\textit{J. A. Oaks}, Indiana Univ. Math. J. 43, No. 3, 959--981 (1994; Zbl 0835.53048)].\ They describe two examples of curves with self-intersections: the symmetric figure-eight and the cardioid, for which exactly one self-intersection of the solution vanishes into the singularity, and they describe their evolution through curve shortening flow. The paper presents and analyzes different examples of solutions where a singularity absorbs more than one self-intersection. The authors define a loop of a solution to curve shortening flow \(\gamma :(t_{0},t_{1})\times…
reviewed by: Alain Brillard
Mathematics Subject Classification ID: 53E10
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35K55
zbMATH DE Number: 7939226
zbMATH Keywords: curve shortening flow
zbMATH Keywords: curve singularities
zbMATH Keywords: knots
zbMATH Keywords: tangles
zbMATH Keywords: loops
zbMATH Keywords: numerical simulation
MaRDI profile type: Publication
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DOI: 10.1088/1361-6544/AD8699
description: scientific article; zbMATH DE number 7939226