Evolving convex curves to constant-width ones by a perimeter-preserving flow
type: Item
instance of: scholarly article
title: Evolving convex curves to constant-width ones by a perimeter-preserving flow
zbMATH Open document ID: 1321.53005
author: Laiyuan Gao
author: Shengliang Pan
published in: Pacific Journal of Mathematics
publication date: 2014-12-19T00:00:00Z
review text: The authors study the flow of closed strictly convex smooth plane curves \(\left\{X(\cdot,t)\right\}_{t\in \left[0,T\right[}\) governed by the system \[ \begin{cases} (\frac{\partial X}{\partial t})(\theta,t)=(w(\theta,t)-\eta(\theta,t))N(\theta,t),&(\theta,t)\in S^{1}\times\left]0,T\right[,\\ X(\theta,0)=X_{0}(\theta),&\theta\in S^{1}, \end{cases}\tag{\(\ast\)} \] where \(X(\cdot,t)\) is parameterised by its tangential angle \(\theta\), \(w(\cdot,t)\) is its width and \(\eta(\theta,t)=\rho(\theta,t)+\rho(\theta+\pi,t)\) with \(\rho(\cdot,t)\) its radius of curvature. The main result is that if the pinching condition \(M<3m\) holds, where \(M=\max\limits_{S^{1}} \rho(\cdot,0)\) and \(m=\min\limits_{S^{1}}\rho(\cdot,0)\), then (\(\ast\)) has a unique global solution \(\{X(\cdot,t)\}_{t\in\left[0,\infty\right[}\) of constant perimeter and increasing area with smooth limit curve \(X_{\infty}:S^{1}\rightarrow\mathbb{R}^{2}\) of constant width, and \(X_{\infty}\) is a circle iff \(X_{0}\) is centrosymmetric. The structure of the paper is as follows: Following a motivation of this flow in Section 1, Section 2 is devoted to reducing the study of (\(\ast\)) to that of a linear Cauchy problem for \(\rho\) and \(\eta\). In Section 3 this reformulation is used together with the pinching condition to establish the existence of a unique global solution to (\(\ast\)). Then the area-increasing and perimeter-preserving properties are verified, and finally the asymptotic behaviour of \(X(\cdot,t)\) is studied, culminating in the convergence to a constant-width curve. Here, it is also shown that \(X(\cdot,t)\) becomes more circular in an appropriate sense as \(t\rightarrow\infty\) and that \(X_{\infty}\) is a circle iff \(X_{0}\) is centrosymmetric, which is in strong contrast to related flows such as that of [\textit{M. Gage}, Contemp. Math. 51, 51--62 (1986; Zbl 0608.53002)] where strictly convex curves necessarily converge to circles. Finally, Section 4 is devoted to examples wher…
reviewed by: Ahmad Afuni
Mathematics Subject Classification ID: 53A04
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 35K15
zbMATH DE Number: 6381867
zbMATH Keywords: convex curves
zbMATH Keywords: curves of constant width
zbMATH Keywords: perimeter-preserving curve flow
MaRDI profile type: Publication
full work available at URL: pjm.2014.272.131
OpenAlex ID: W2013434837
DOI: 10.2140/PJM.2014.272.131
description: scientific article; zbMATH DE number 6381867