On Yau's problem of evolving one curve to another: convex case
type: Item
instance of: scholarly article
title: On Yau's problem of evolving one curve to another: convex case
zbMATH Open document ID: 1403.53004
author: Laiyuan Gao
author: Yun-tao Zhang
published in: Journal of Differential Equations
publication date: 2018-11-06T00:00:00Z
review text: Let \(X_0\) and \(\tilde{X}\) be two smooth, closed and embedded curves with positive curvature everywhere. In this paper, the authors consider a revised Yau curvature difference flow: \[ \begin{cases} \frac{\partial X}{\partial t}(\varphi,t)=[\kappa(\varphi,t)- \lambda(t)\tilde{\kappa}(\varphi)]N(\varphi,t)\;\text{ in }\;S^1\times(0,\omega),\\ X(\varphi,0)=X_0(\varphi)\;\text{ on }\;S^1, \end{cases} \] where \(\kappa\) and \(\tilde{\kappa}\) are the curvature of the evolving curve \(X\) and the target curve \(\tilde{X}\) separately, \(N\) is the unit inner normal of \(X\), \(\lambda(t)=2\pi/\int_{S^1}\tilde{\kappa}gd\varphi\) and \(g\) is the metric of \(X\). They prove that this flow exists globally on time interval \([0,+\infty)\), preserves the bounded area \(A\) and the convexity of every evolving curve \(X(\cdot,t)\) and deforms \(X_0\) into a fixed curve \(X_{\infty}\) (congruent to \(\sqrt{A/\tilde{A}}\tilde{X}\)) as time tends to infinity, where \(\tilde{A}\) is the area bounded by the curve \(\tilde{X}\).
reviewed by: Atsushi Fujioka
Mathematics Subject Classification ID: 53A04
Mathematics Subject Classification ID: 35K15
zbMATH DE Number: 6973891
zbMATH Keywords: convex curve
zbMATH Keywords: Yau's curvature difference flow
zbMATH Keywords: \(C^{\infty}\) convergence
zbMATH Keywords: Steiner point
MaRDI profile type: Publication
full work available at URL: j.jde.2018.07.037
OpenAlex ID: W2883283372
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Wikidata QID: Q129458772
DOI: 10.1016/J.JDE.2018.07.037
description: scientific article; zbMATH DE number 6973891