The affine shape of a figure 8 under the curve shortening flow
type: Item
instance of: scholarly article
title: The affine shape of a figure 8 under the curve shortening flow
author: Matei P. Coiculescu
author: Richard Evan Schwartz
published in: Journal of Differential Geometry
publication date: 2024-08-27T00:00:00Z
review text: The paper under review advances the theory of geometric flows and deals with the curve shortening flow for closed curves in the Euclidean plane \(\mathbb R^2\).\N\NBy definition, the curve shortening flow is represented by a smooth family of closed curves \(C: S^1\times [0,T)\to \mathbb R^2\) whose evolution in time \(t\) is governed by the equation \(\frac{dC}{dt} = k N\), where \(k\) and \(N\) are the curvature and the unit normal of the evolving curve, respectively.\N\NIt is well-known that if an initial closed curve \(C_0=C(\cdot, 0)\) is embedded, then under the flow it shrinks to a ``round'' point in a finite time \(T\), see [\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)].\N\NOn the other hand, if \(C_0\) has self-intersections then its evolution under the flow turns out to be more complicated. Particularly, if \(C_0\) is a figure-eight curve, a closed curve with one point of self-intersection, then either of the following two cases can happen: \N(i) if the two lobes of \(C_0\) have different areas, then the curve evolves so that one lobe shrinks to a point before the vanishing time \(T\), the flow expands through this singularity, and then the evolving curve shrinks to a point; \N(ii) if the lobes of \(C_0\) have the same area, then the double point doesn't disappear before the vanishing time \(T\) and the isoperimetric ratio of the evolving curve grows to \(\infty\) as \(t\to T\), see [\textit{M. A. Grayson}, Invent. Math. 96, No. 1, 177--180 (1989; Zbl 0669.53003)]. \NIt is conjectured that in the case (ii) the shrinking to a point holds true as well. The conjecture is confirmed in some particular cases only, but in general the problem remains unsolved.\N\NInspired by the Grayson conjecture, the authors explore the curve shortening flow for a particular family of figure-eight curves. Namely, a figure-eight curve \(C: S^1\to \mathb…
reviewed by: Vasyl Gorkaviy
Mathematics Subject Classification ID: 53E10
Mathematics Subject Classification ID: 53A04
Mathematics Subject Classification ID: 58J35
zbMATH DE Number: 7902274
zbMATH Keywords: curve shortening flow
zbMATH Keywords: figure-eight curve
zbMATH Keywords: grim reaper solitons
zbMATH Keywords: maximum principle
zbMATH Keywords: curvature blowup
MaRDI profile type: Publication
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DOI: 10.4310/JDG/1721071494
description: scientific article; zbMATH DE number 7902274