Shortening embedded curves
type: Item
instance of: scholarly article
title: Shortening embedded curves
zbMATH Open document ID: 0686.53036
DOI: 10.2307/1971486
author: Matthew A. Grayson
published in: Annals of Mathematics. Second Series
publication date: 1989-01-01T00:00:00Z
review text: Let F be a surface. We consider the evolution equation for a map C(u,t): \(S^ 1\times 1\to F\) given by \[ (*)\quad \partial C/\partial t=kN,\quad C(u,0)=c(u), \] where k is the curvature of C and N is its unit normal vector. Now suppose F is convex at infinity, namely the convex hull of every compact subset is compact. It is proved that there exists a solution for (*) on \([0,t_{\infty})\). If \(t_{\infty}\) is finite, then C converges to a point. If \(t_{\infty}\) is infinite, then the curvature of C converges to zero in the \(C^{\infty}\) norm. As an application, it is proved that a 2-sphere with a smooth Riemannian metric has at least three simple closed geodesics.
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 58J35
zbMATH DE Number: 4124527
zbMATH Keywords: evolution of curvature curve shortening
zbMATH Keywords: evolution equation
Wikidata QID: Q57535365
MaRDI profile type: Publication
OpenAlex ID: W2333300544
description: scientific article; zbMATH DE number 4124527