A density function and the structure of singularities of the mean curvature flow
type: Item
instance of: scholarly article
title: A density function and the structure of singularities of the mean curvature flow
zbMATH Open document ID: 0833.35062
DOI: 10.1007/BF01192093
author: Andrew Stone
published in: Calculus of Variations and Partial Differential Equations
publication date: 1996-03-18T00:00:00Z
review text: The author studies the formation of singularities in the mean curvature flow \({d\over dt} F(p, t)= H(p, t) \nu(p, t)\), where \(F(\cdot, t)\) is a family of smooth embeddings of a closed hypersurface in \(\mathbb{R}^{n+ 1}\), \(\nu(\cdot, t)\) the inward pointing unit normal and \(H(\cdot, t)\) the corresponding mean curvature at ``time'' \(t\). The results of this paper extend and refine previous ones of G. Huisken. Denoting by \(A\) the second fundamental form of a family of hypersurfaces as above one considers the first time \(T\) when \(\max_p|A|^2(p, t)\) becomes infinite. By the results of Huisken, such finite \(T\) exists and \(\max_p|A|^2(p, t)\geq {1\over 2}(T- t)^{- 1}\). As a fundamental hypothesis the converse estimate \(\max_p|A|^2(p, t)\leq {1\over 2} C_0(T- t)^{- 1}\) for some constant \(C_0\) is assumed throughout the paper; it is known to hold in many, though not all cases. For flow starting from a hypersurface with \(H\geq 0\) (and therefore staying in this class) the author's first result reads as follows: Whenever \(p_k\to p\) and \(t_k\uparrow T\) \((k\to \infty)\) such that \(|A|^2(p_k, t_k)\geq \delta(T- t_k)^{- 1}\) for some \(\delta> 0\) then already \(\limsup_{t\to T} (T- t)|A|^2(p, t)> 0\), which puts one in a situation analyzed before by G. Huisken: From any sequence \(t_k\to T\) one can extract a convergent subsequence (again denoted by \(t_k\)) such that the images of \((2(T- t_k))^{- 1/2}(F(\cdot, t_k)- \widehat p)\), \(\widehat p:= \lim_{t\to T} F(p, T)\), converge to a cylinder with cross section some \(m\)-dimensional round sphere, \(1\leq m\leq n\). A further result of the present author then states that the various blow up limits with respect to a fixed singular point \(\widehat p\) can only differ by a rotation of \(\mathbb{R}^{n+ 1}\). Furthermore, the author addresses the question whether blow up of \(|A|^2\) can occur with a slower rate than considered above and he proves curvature estimates which show e.g. that if \(|A|^2\) …
Mathematics Subject Classification ID: 35G10
Mathematics Subject Classification ID: 35K25
Mathematics Subject Classification ID: 58J35
zbMATH DE Number: 720499
zbMATH Keywords: formation of singularities
zbMATH Keywords: mean curvature flow
zbMATH Keywords: blow up
reviewed by: F. Tomi
MaRDI profile type: Publication
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Wikidata QID: Q125825165
description: scientific article; zbMATH DE number 720499