Asymptotic behavior of anisotropic curve flows
type: Item
instance of: scholarly article
title: Asymptotic behavior of anisotropic curve flows
zbMATH Open document ID: 0917.53014
published in: Journal of Differential Geometry
publication date: 1999-07-29T00:00:00Z
review text: In this paper, the asymptotic behavior of a closed embedded curve evolving by an anisotropic parabolic equation \(\gamma_t = (\Phi(T)k + \Psi(T))N\) on a Riemannian surface is studied (\(T\), \(N\) are the unit, respectively normal vectors of \(\gamma\), \(k\) denotes its curvature, and the functions \(\Phi\) and \(\Psi\) are rather arbitrary and have to obey some general estimates and restrictions only). Whenever the evolving curve develops a singularity, it shrinks to a point [\textit{J. A. Oaks}, Indiana Univ. Math. J. 43, 959-981 (1994; Zbl 0835.53048)]. The main result in this paper is that the rescaled evolving curve converges to a Minkowski isoperimetrix associated to \(\Phi\) at the shrinking point of \(\gamma_t\) in the \(C^{\infty}\) topology. For the Euclidean case, cf. \textit{K. Chou} and \textit{X. Zhu}, Duke Math. J. 97, 579-619 (1999)].
Mathematics Subject Classification ID: 53C21
Mathematics Subject Classification ID: 37C10
Mathematics Subject Classification ID: 53A04
Mathematics Subject Classification ID: 35G10
Mathematics Subject Classification ID: 35K25
Mathematics Subject Classification ID: 58J37
zbMATH DE Number: 1203421
zbMATH Keywords: anisotropic flow
zbMATH Keywords: evolution of curves
zbMATH Keywords: curves on Riemannian surfaces
zbMATH Keywords: Minkowski isoperimetrix
zbMATH Keywords: isoperimetric estimates
zbMATH Keywords: singularities
author: Xi-Ping Zhu
reviewed by: Eberhard Teufel
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1214460796
OpenAlex ID: W1521362484
Wikidata QID: Q115174766
DOI: 10.4310/JDG/1214460796
description: scientific article; zbMATH DE number 1203421