Mean curvature flow with surgeries of two-convex hypersurfaces
type: Item
instance of: scholarly article
title: Mean curvature flow with surgeries of two-convex hypersurfaces
zbMATH Open document ID: 1170.53042
author: Gerhard Huisken
author: Carlo Sinestrari
published in: Inventiones Mathematicae
publication date: 2009-03-20T00:00:00Z
review text: Inspired by Hamilton's Ricci flow with surgery, the authors define a way to extend the mean curvature flow beyond singularities. The idea is to stop the flow close to the singularity time and replace the regions with high curvature by more regular ones, dropping in fact the maximum curvature by a fixed factor. Simultaneously, one also removes the connected components of the hypersurface which are diffeomorphic to \({\mathbb{S}}^n\) or \({\mathbb{S}}^{n-1} \times {\mathbb{S}}^1\). The main theorem shows that for an immersed smooth \(2\)-convex hypersurface of dimension three or greater, the mean curvature flow with surgery exists and terminates in a finite number of steps when the hypersurface is diffeomorphic to either \({\mathbb{S}}^n\) or to a finite connected sum of \({\mathbb{S}}^{n-1} \times {\mathbb{S}}^1\). Very rigourously, the paper treats convexity estimates under surgery, the replacement of necks, geometric cylinders, by two spherical caps and a needed gradient estimate. The latter differs from previous estimates through the fact that the constants do not depend on the maximum of the curvature on the hypersurface, despite its pointwise character. The existence of an algorithm describing the mean curvature with surgery, as well as the finite number of steps of the algorithm concluded by the main result, combine very technical parts.
reviewed by: Alina Stancu
Mathematics Subject Classification ID: 53C44
Mathematics Subject Classification ID: 53C42
zbMATH DE Number: 5530518
zbMATH Keywords: mean curvature flow
zbMATH Keywords: neck surgery
zbMATH Keywords: singularities
zbMATH Keywords: weak convexity
MaRDI profile type: Publication
full work available at URL: s00222-008-0148-4
OpenAlex ID: W2015932201
cites work: Mean curvature flow through singularities for surfaces of rotation
cites work: Q5690183
cites work: The Motion of a Surface by Its Mean Curvature. (MN-20)
cites work: Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations
cites work: Singularity structure in mean curvature flow of mean-convex sets
cites work: Regularity theory for mean curvature flow
cites work: Motion of level sets by mean curvature. I
cites work: Minimal disks and two-convex hypersurfaces
cites work: Four-manifolds with positive curvature operator
cites work: Convex hypersurfaces with pinched second fundamental form
cites work: Harnack estimate for the mean curvature flow
cites work: Q4873981
cites work: Four-manifolds with positive isotropic curvature
cites work: Flow by mean curvature of convex surfaces into spheres
cites work: Q3137116
cites work: Mean curvature flow singularities for mean convex surfaces
cites work: Convexity estimates for mean curvature flow and singularities of mean convex surfaces
cites work: Q3905599
cites work: Local rigidity theorems for minimal hypersurfaces
cites work: An introduction to Morse theory. Transl. from the Japanese by Kiki Hudson and Masahico Saito
cites work: Low codimensional submanifolds of Euclidean space with nonnegative isotropic curvature
cites work: Sobolev and mean‐value inequalities on generalized submanifolds of Rn
cites work: Q5485637
cites work: Q5485638
cites work: P-convex Riemannian manifolds
cites work: The topology of hypersurfaces moving by mean curvature
cites work: The nature of singularities in mean curvature flow of mean-convex sets
cites work: Q3780060
Wikidata QID: Q125724937
DOI: 10.1007/S00222-008-0148-4
description: scientific article; zbMATH DE number 5530518