Mean curvature flow with surgeries of two–convex hypersurfaces

By G. Huisken, Carlo Sinestrari. 2009

210 citations recorded by the source corpus.

We consider a closed smooth hypersurface immersed in euclidean space evolving by mean curvature flow. It is well known that the solution exists up to a finite singular time at which the curvature becomes unbounded. The purpose of this paper is to define a flow after singularities by a new approach based on a surgery procedure. Compared with the notions of weak solutions existing in the literature, the flow with surgeries has the advantage that it keeps track of the changes of topology of the evolving surface and thus can be applied to classify all geometries that are possible for the initial manifold. Our construction is inspired by the procedure originally introduced by Hamilton for the Ricci flow, and then employed by Perelman in the proof of Thurston's geometrization conjecture. In this paper we consider initial hypersurfaces which have dimension at least three and are two-convex, that is, such that the sum of the two smallest principal curvatures is nonnegative everywhere. Under these assumptions, we construct a flow with surgeries which has uniformly bounded curvature until the evolving manifold is split in finitely many components with known topology. As a corollary, we obtain a classification up to diffeomorphism of the hypersurfaces under consideration.

Subfields: Applied Mathematics, Astronomy and Astrophysics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/9a291afb6a3c8edee3f0d98fb90078e99bf60944

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Citations in the corpus

Uniqueness of the bowl soliton (2014, background): “Uniform 2–convexity is preserved under mean curvature flow, and arises eg in the construction of mean curvature flow with surgery by Huisken and Sinestrari [17]; see also Haslhofer and Kleiner [13] and Brendle and Huisken [6].”
Mean Curvature Flow of Mean Convex Hypersurfaces (2013, methodology): “As an immediate consequence of (the smooth admissible version of) the convexity estimate, we obtain the original versions of the convexity estimates due to Huisken and Sinestrari [25, theorem 1.4] and White [38, theorem 8]: COROLLARY 2.10.”
Convex ancient solutions of the mean curvature flow (2014, methodology): “1 in [19], without the constant remainder term, provided k is small enough compared to n.”
Mean curvature flow with surgery (2014, methodology): “In September 2013, Brendle and Huisken posted a very interesting preprint [BH13], where they also solved the case N = 3, by combining the work of Huisken-Sinestrari [HS09], the local curvature estimate from our previous paper [HK13a], and Brendle’s estimate for the inscribed radius [Bre13a].”
Ancient asymptotically cylindrical flows and applications (2019, methodology): “In fact, a coarser description of the behavior of such flows at high curvature regions turned out to be sufficient for the construction of mean curvature flow with surgery by Huisken-Sinestrari [HS09], Brendle-Huisken [BH16] and Haslhofer-Kleiner [HK17b].”
Mean Curvature Flow in Higher Codimension: Introduction and Survey (2011, background): “In [54] Huisken and Sinestrari define such a notion based on a surgery procedure.”
Uniqueness of two-convex closed ancient solutions to the mean curvature flow (2018, methodology): “By the cylindrical estimate ([13], [12]) we have that for every > 0 there exists a j0 so that for j ≥ j0.”
The singular set of mean curvature flow with generic singularities (2014, background): “Moreover, if the initial hypersurface is mean convex, then all singularities are cylindrical; see [3,27,30–33,48,49,53].”
Mean curvature flow with surgery of mean convex surfaces in R3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb (2013, methodology): “Proposition 2.12 (Huisken-Sinestrari [16], Section 4).”
A sharp bound for the inscribed radius under mean curvature flow (2013, background): “This last step closely follows the work of Huisken [11] (see also [14], Section 4).”
Ricci flow with surgery on manifolds with positive isotropic curvature (2017, background): “Moreover, in the two-convex setting, Huisken and Sinestrari [23] showed that mean curvature flow can be extended beyond singularities by a surgery procedure.”