Flow by mean curvature of convex surfaces into spheres
type: Item
instance of: scholarly article
title: Flow by mean curvature of convex surfaces into spheres
zbMATH Open document ID: 0556.53001
author: Gerhard Huisken
published in: Journal of Differential Geometry
publication date: 1984-01-01T00:00:00Z
review text: Let \(M_ 0\) be a smooth closed convex hypersurface with everywhere positive curvatures in Euclidean space \({\mathbb{R}}^{n+1}\). Suppose that \(M_ 0\) is smoothly deformed (that is, embedded into a smooth family \((M_ t\), \(t\geq 0)\) of hypersurfaces depending on a time parameter) such that, at every time t, the rate of change of \(M_ t\) in direction of the inward unit normal vector is equal to the (positive) mean curvature of the hypersurface at the point considered. The author shows that the parabolic evolution equation describing the problem has a smooth solution on a finite time interval \(0\leq t\leq T\), and the \(M_ t's\) converge to a point as \(t\to T\). Moreover, if the surfaces undergo suitable homotheties and the time parameter is transformed appropriately into a parameter \(\tilde t,\) \(0\leq \tilde t<\infty\), it is shown that the normalized surfaces converge to a sphere in the \(C^{\infty}\)-topology as \(\tilde t\to \infty\). As the author says, his approach is inspired by a paper of \textit{R. S. Hamilton} [J. Differ. Geom. 17, 255-306 (1982; Zbl 0504.53034)], and he can use many of the methods developed there. The case \(n=1\) was treated by \textit{M. E. Gage} [Invent. Math. 76, 357-364 (1984; Zbl 0542.53004)]. \{Reviewer's remark: A similar problem, related in spirit though not in the methods, was treated by \textit{W. J. Firey} [Mathematika 21, 1-11 (1974; Zbl 0311.52003)]. There the rate of change of the support function is proportional to the Gauss curvature.\}
reviewed by: Rolf Schneider
Mathematics Subject Classification ID: 53A07
Mathematics Subject Classification ID: 35G10
Mathematics Subject Classification ID: 35K25
Mathematics Subject Classification ID: 52A20
zbMATH DE Number: 3886661
zbMATH Keywords: convex hypersurface
zbMATH Keywords: mean curvature
zbMATH Keywords: evolution equation
Wikidata QID: Q115187557
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1214438998
OpenAlex ID: W1491650242
DOI: 10.4310/JDG/1214438998
description: scientific article; zbMATH DE number 3886661