The heat equation shrinks embedded plane curves to round points
type: Item
instance of: scholarly article
title: The heat equation shrinks embedded plane curves to round points
zbMATH Open document ID: 0667.53001
author: Matthew A. Grayson
published in: Journal of Differential Geometry
publication date: 1987-01-01T00:00:00Z
review text: This paper contains the final solution of the long-standing ``curve- shortening problem'' for plane curves: Let \(\gamma_ 0: S^ 1\to {\mathbb{R}}^ 2\) be a regular embedded closed plane curve. Then the evolution equation \({\dot \gamma}=k\cdot N\) (N a unit normal field, k the curvature) with initial condition \(\gamma (0,s)=\gamma_ 0(s)\) always has a solution \(\gamma: {\mathbb{R}}^+\times S^ 1\to {\mathbb{R}}^ 2,\) \(S\mapsto \gamma_ t(s)=\gamma (t,s)\) is an embedded curve for all t and \(\gamma_ t\) approaches a (shrinking) round circle as \(t\to \infty\).
Mathematics Subject Classification ID: 53A04
Mathematics Subject Classification ID: 35G10
Mathematics Subject Classification ID: 35K25
zbMATH DE Number: 4092248
zbMATH Keywords: curve-shortening problem
zbMATH Keywords: closed plane curve
zbMATH Keywords: evolution equation
reviewed by: Ulrich Pinkall
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1214441371
OpenAlex ID: W1498500832
Wikidata QID: Q57535361
DOI: 10.4310/JDG/1214441371
description: scientific article; zbMATH DE number 4092248