Regularity theory for mean curvature flow
type: Item
instance of: scholarly article
title: Regularity theory for mean curvature flow
zbMATH Open document ID: 1058.53054
author: Klaus Ecker
published in: Progress in Nonlinear Differential Equations and Their Applications
publication date: 2004-03-24T00:00:00Z
review text: This book is devoted to the regularity theory for mean curvature flow. The mean curvature flow evolves hypersurfaces in their normal direction with speed equal to the mean curvature, it is a descent flow for the area functional. An evolving hypersurface may become non-regular in a finite time. The regularity theory for mean curvature flow consists of results regarding the set of singular times, the dimension and structure of the singular set, the asymptotic behaviour of solutions near singularities, etc. For example, for any regular compact convex initial hypersurface \(M_0\) the solution of the mean curvature flow, \(M_t\), remains regular, compact and convex until it disappears into a round point in finite time \(t_0\) (like a family of spheres shrinking to a point). The author's aim is to present some recent results related to Brakke's main regularity theorem: for the special case where smooth solutions develop singularities for the first time, this theorem states that under certain additional assumptions the hypersurfaces are still smooth at the singular time except for a lower dimensional set. In order to clarify and simplify main ideas, the author replaces Brakke's original techniques with use of geometric measure theory by more recent methods from differential geometry and PDE theory. The covered topics include original point-wise estimates on geometric quantities (the position vector and its derivatives, the curvature and its derivatives, etc.) for hypersurfaces evolving by mean curvature, area estimates (first variation formula, area decay, mean curvature flow in integral form, mean value inequality, etc) and Huisken's monotonicity formula, recent developments by B. White, T. Ilmanen and K. Ecker where Brakke's theorem has been improved. The main part of the book is followed by an appendix consisting of 6 chapters, where the author presents in more detail some background, provides an account of fundamental techniques in minimal surface theory and discusses …
Mathematics Subject Classification ID: 53C44
Mathematics Subject Classification ID: 53-02
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 53C21
zbMATH DE Number: 2061766
zbMATH Keywords: mean curvature formula
zbMATH Keywords: Brakke's regularity theorem, monotonicity formula
zbMATH Keywords: mean value formula
reviewed by: Vasyl Gorkaviy
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 2061766