A distance comparison principle for evolving curves
type: Item
instance of: scholarly article
title: A distance comparison principle for evolving curves
zbMATH Open document ID: 0931.53032
author: Gerhard Huisken
published in: The Asian Journal of Mathematics
publication date: 2000-02-20T00:00:00Z
review text: The author investigates the quotient of extrinsic distance \(d=|\gamma(p)-\gamma(q)|\) and intrinsic distance \(l=\int_p^qds\) for points \(p, q\) on an embedded curve \(\gamma\) in the plane evolving by the curve shortening flow. In the case of closed curves, \(l\) is replaced by \((L/\pi)\sin(l\pi/L)\) where \(L\) is the length of \(\gamma\). The main result is that the ratio \(d/l\), considered as a measure for the straightness of an embedded curve, improves at a local minimum. As a direct application of this result, the author gives an elegant proof for Grayson's convergence result [\textit{M. Grayson}, J. Differ. Geom. 26, 285-314 (1987; Zbl 0667.53001)] where he makes use of the well known fact that singularities in the curve shortening flow equal, after an appropriate rescaling, a selfsimilar evolving curve or a translating soliton, the ``grim reaper curve'' given by \(y=t-\log\cos x\). The latter cannot appear since the ratio between extrinsic and intrinsic distance becomes arbitrarily small, contradicting the main theorem in this paper. Now Grayson's result follows since the only embedded selfsimilar curve is the round circle. In the remainder, some applications to nonclosed curves with boundaries are given.
reviewed by: Knut Smoczyk
Mathematics Subject Classification ID: 53C44
Mathematics Subject Classification ID: 53A04
zbMATH DE Number: 1240915
zbMATH Keywords: curve shortening flow
zbMATH Keywords: selfsimilar curve
Wikidata QID: Q57535373
MaRDI profile type: Publication
DOI: 10.4310/AJM.1998.V2.N1.A2
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description: scientific article; zbMATH DE number 1240915