On regularity for mean curvature flow of hypersurfaces
type: Item
instance of: scholarly article
title: On regularity for mean curvature flow of hypersurfaces
zbMATH Open document ID: 0814.53004
DOI: 10.1007/BF01190894
author: Klaus Ecker
published in: Calculus of Variations and Partial Differential Equations
publication date: 1995-02-06T00:00:00Z
review text: A family of isometric immersions \(x_ t : M^ n \to \mathbb{R}^{n+1}\) of a parameter manifold \(M^ n\) into the Euclidean space \(\mathbb{R}^{n+1}\) is said to move by mean curvature if \({d \over dt} x_ t\) equals the mean curvature vector. It is known that the hypersurfaces \(M_ t = x_ t (M^ n)\) can become singular after a finite time interval \((0,T)\). The author surveys results on singularities or on regularity of his own as well as of others. The heat operator \({\partial \over \partial t} - \Delta\) of \(\mathbb{R}^{n+1}\) and the square \(| A |^ 2\) of the second fundamental form \(A = A_ t\) of \(M_ t\) are used to derive a priori estimates for the mean curvature flow. For instance, the supremum of \(| A |^ 2\) in some domain is bounded by the mean value of \(| A |^ 2\) in a somewhat bigger domain. Another remarkable result is that, under plausible assumptions, the singular set has Hausdorff measure zero.
Mathematics Subject Classification ID: 53A07
Mathematics Subject Classification ID: 37C10
Mathematics Subject Classification ID: 53-02
zbMATH DE Number: 719390
zbMATH Keywords: singularities
zbMATH Keywords: regularity
zbMATH Keywords: mean curvature flow
reviewed by: Rainer Schimming
MaRDI profile type: Publication
Wikidata QID: Q125119231
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description: scientific article; zbMATH DE number 719390