Paper · Jul 27, 2016

Neckpinch singularities in fractional mean curvature flows

Eleonora Cinti, Carlo Sinestrari, Enrico Valdinoci

math.DGWider flowsarXiv:1607.08032v2
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Abstract

In this paper we consider the evolution of sets by a fractional mean curvature flow. Our main result states that for any dimension , there exists an embedded surface in evolving by fractional mean curvature flow, which developes a singularity before it can shrink to a point. When this result generalizes the analogue result of Grayson for the classical mean curvature flow. Interestingly, when , our result provides instead a counterexample in the nonlocal framework to the well known Grayson Theorem, which states that any smooth embedded curve in the plane evolving by (classical) MCF shrinks to a point.

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Submitted
Jul 27, 2016
Last revised
Jul 28, 2016 (v2)
Subjects
math.DG
Mean curvature flowAnisotropic & nonlocal flowsSingularities

Its neighbourhood

Connected papers

The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2016, plus the canon.

Connected papers This paper in the centre and the 8 papers closest to it around it, in date order clockwise from the top. The closer a paper sits to the centre, the stronger the connection. This paper Some results on anisotropic fractional mean curvature flows (Antonin Chambolle, Matteo Novaga and Berardo Ruffini, 2016)4Chambolle 2016 Fattening and nonfattening phenomena for planar nonlocal curvature flows (Annalisa Cesaroni, Serena Dipierro, Matteo Novaga et al., 2018)2Cesaroni 2018 Convex sets evolving by volume preserving fractional mean curvature flows (Eleonora Cinti, Carlo Sinestrari and Enrico Valdinoci, 2018)1Cinti 2018 Eventual Regularization of Fractional Mean Curvature Flow (Stephen Cameron, 2019)7Cameron 2019 Short time existence of the classical solution to the fractional Mean curvature flow (Vesa Julin and Domenico La Manna, 2019)3Julin 2019 Fractional mean curvature flow of Lipschitz graphs (Annalisa Cesaroni and Matteo Novaga, 2021)6Cesaroni 2021 Stability of the ball under volume preserving fractional mean curvature flow (Annalisa Cesaroni and Matteo Novaga, 2022)8Cesaroni 2022 Convergence of the volume preserving fractional mean curvature flow for convex sets (Vesa Julin and Domenico Angelo La Manna, 2023)5Julin 2023
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Convex sets evolving by volume preserving fractional mean curvature flows

    Eleonora Cinti, Carlo Sinestrari and Enrico ValdinociWider flows

    Topics Anisotropic & nonlocal flows · Same authors Eleonora Cinti, Carlo Sinestrari, Enrico Valdinoci · Shared terms “fractional mean”

  2. Fattening and nonfattening phenomena for planar nonlocal curvature flows

    Annalisa Cesaroni, Serena Dipierro, Matteo Novaga et al.Wider flows

    Topics Anisotropic & nonlocal flows, Singularities · Same author Enrico Valdinoci · Shared terms “fractional mean”

  3. Short time existence of the classical solution to the fractional Mean curvature flow

    Vesa Julin and Domenico La MannaWider flows

    Topics Anisotropic & nonlocal flows · Shared terms “fractional mean”

  4. Some results on anisotropic fractional mean curvature flows

    Antonin Chambolle, Matteo Novaga and Berardo RuffiniWider flows

    Topics Anisotropic & nonlocal flows · Shared terms “fractional mean”

  5. Convergence of the volume preserving fractional mean curvature flow for convex sets

    Vesa Julin and Domenico Angelo La MannaWider flows

    Topics Anisotropic & nonlocal flows, Singularities · Shared terms “fractional mean”

  6. Fractional mean curvature flow of Lipschitz graphs

    Annalisa Cesaroni and Matteo NovagaWider flows

    Topics Anisotropic & nonlocal flows · Shared terms “fractional mean”

  7. Eventual Regularization of Fractional Mean Curvature Flow

    Stephen CameronWider flows

    Topics Anisotropic & nonlocal flows · Shared terms “fractional mean”, “classical mean”

  8. Stability of the ball under volume preserving fractional mean curvature flow

    Annalisa Cesaroni and Matteo NovagaWider flows

    Topics Anisotropic & nonlocal flows · Shared terms “fractional mean”

Read around it

Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Mean curvature flow Wider flows A hypersurface moves in the direction of its mean curvature vector, which decreases area as fast as possible in the L² sense. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) Anisotropic & nonlocal flows Wider flows When surface energy depends on direction, as in crystals, mean curvature flow becomes anisotropic, and the Wulff shape replaces the round sphere. Start with: Crystalline variational problems (Taylor, 1978) Singularities On a closed manifold the flow can stop in finite time: curvature blows up, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993)

In the guidebook

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  2. 6C.8
    Monotonicity, ε-Regularity and Tangent ConesBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Monotonicity formulas for minimal surfaces and harmonic maps, small energy implies smoothness, bubble trees, tangent cones and dimension reduction, with soap films, liquid crystals and pseudolocality.

  3. 11B.4
    Blow-Ups, Blow-Downs and Splitting at InfinityBook 11B · Solitons, Compactness and Singularities

    Limits of rescaled manifolds and flows: why blow-up limits are nonnegatively curved, what blow-downs see, and how lines appear at infinity.