The heat equation shrinking convex plane curves
type: Item
instance of: scholarly article
title: The heat equation shrinking convex plane curves
zbMATH Open document ID: 0621.53001
author: M. Gage
author: Richard S. Hamilton
published in: Journal of Differential Geometry
publication date: 1986-01-01T00:00:00Z
review text: Given a curve in the plane, let it deform in the direction of its curvature vector. It was conjectured that the curve would collapse in finite time to a point and that, as the curve collapsed, it would become asymptotically close to a shrinking standard circle. The authors prove this conjecture for a convex plane curve. In a more recent paper [J. Differ. Geom. 26, 285-314 (1987)], \textit{M. A. Grayson} shows that any embedded curve eventually becomes convex, completely settling the conjecture. Integral estimates (and the isoperimetric inequality) are used to obtain the necessary bounds for proving long time existence and convergence towards a shrinking circle. This is in contrast to G. Huisken's work on deforming higher dimensional hypersurfaces by the mean curvature vector, where the Codazzi-Mainardi equations and the maximum principle are used.
Mathematics Subject Classification ID: 53A04
Mathematics Subject Classification ID: 58J35
Mathematics Subject Classification ID: 58D10
zbMATH DE Number: 4007132
zbMATH Keywords: heat equation
zbMATH Keywords: curvature vector
zbMATH Keywords: convex plane curve
zbMATH Keywords: shrinking circle
Wikidata QID: Q57535360
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1214439902
OpenAlex ID: W1553761880
DOI: 10.4310/JDG/1214439902
description: scientific article; zbMATH DE number 4007132