Convexity estimates for mean curvature flow and singularities of mean convex surfaces
type: Item
instance of: scholarly article
title: Convexity estimates for mean curvature flow and singularities of mean convex surfaces
zbMATH Open document ID: 0992.53051
DOI: 10.1007/BF02392946
author: Gerhard Huisken
author: Carlo Sinestrari
published in: Acta Mathematica
publication date: 2000-12-06T00:00:00Z
review text: In this nice paper, the authors obtain the following classification of singularities which form in hypersurfaces with nonnegative mean curvature moving under mean curvature flow: Consider a smooth immersion of a closed, \(n\)-dimensional hypersurface with nonnegative mean curvature, which evolves by mean curvature flow on a finite maximal time interval \([0,T).\) Then any rescaled limit of a singularity that forms as \(t \rightarrow T\) is weakly convex. This implies that for a hypersurface of nonnegative mean curvature, the limiting flow of a Type-II singularity has convex surfaces \(\widetilde M_{\tau},\) \(\tau \in \mathbb R\). Moreover, \(\widetilde M_{\tau}\) is either a strictly convex translating soliton, or it splits as a product of \(\mathbb R^{n-m}\) with a lower-dimensional strictly convex translating soliton in \(\mathbb R^{m+1}.\) This classification complements that of Type-I singularities [\textit{G. Huisken}, J. Differ. Geom. 31, 285-299 (1990; Zbl 0694.53005); Proc. Sympos. Pure Math. 54, Part 1, 175-191 (1993; Zbl 0791.58090)]. To prove the result, the authors derive a priori bounds on the symmetric functions of the principal curvatures of the evolving hypersurfaces. More precisely, let \((\lambda_{1}, \ldots,\lambda_{n})\) be the principle curvatures, and let \(S_{k}(\lambda)=\sum_{1\leq i_{1}<i_{2}\ldots<i_{k}\leq n} \lambda_{i_{1}}\ldots\lambda_{i_{k}}\). Then for each \(k,\) \( 2\leq k \leq n\), and any \(\eta>0\), there exists a constant \(C\) depending only on \(\eta, k, n,\) and the initial data, so that \[ S_{k}(\lambda) \geq -\eta H^{k} - C_{\eta,k}. \] The idea of the derivation is to proceed inductively on the degree \(k\) of the symmetric functions, with the \(k=2\) case contained in a previous paper of the authors [Calc. Var. Partial Differ. Equ. 8, 1-14 (1999; Zbl 0992.53052)]. A crucial step is to perturb the second fundamental form in such a way as to be able to work with the quotient of the new consecutive symmetric functions as a …
reviewed by: Christine Guenther
Mathematics Subject Classification ID: 53C44
Mathematics Subject Classification ID: 35K55
zbMATH DE Number: 1541212
zbMATH Keywords: mean curvature flow
zbMATH Keywords: singularities
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02392946
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description: scientific article; zbMATH DE number 1541212