Paper · May 22, 2019

Eventual Regularization of Fractional Mean Curvature Flow

Stephen Cameron

math.APWider flowsarXiv:1905.09184
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Abstract

We show that any open set that is a finite distance away from a Lipschitz subgraph will become a Lipschitz subgraph after flowing under fractional mean curvature flow for a finite, universal time. Our proof is quantitative and inherently nonlocal, as the corresponding statement is false for classical mean curvature flow. This is the first regularizing effect proven for weak solutions to nonlocal curvature flow.

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Submitted
May 22, 2019
Subjects
math.AP

Comments: 44 pages, 7 figures

Mean curvature flowOther curvature flowsAnisotropic & nonlocal flowsWeak & singular flows

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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)7Chambolle 2016 Neckpinch singularities in fractional mean curvature flows (Eleonora Cinti, Carlo Sinestrari and Enrico Valdinoci, 2016)3Cinti 2016 Convex sets evolving by volume preserving fractional mean curvature flows (Eleonora Cinti, Carlo Sinestrari and Enrico Valdinoci, 2018)8Cinti 2018 Short time existence of the classical solution to the fractional Mean curvature flow (Vesa Julin and Domenico La Manna, 2019)5Julin 2019 Stability results for nonlocal geometric evolutions and limit cases for fractional mean curvature flows (Annalisa Cesaroni, Lucia De Luca, Matteo Novaga et al., 2020)1Cesaroni 2020 Convergence of supercritical fractional flows to the mean curvature flow (Lucia De Luca, Andrea Kubin and Marcello Ponsiglione, 2021)6De Luca 2021 Fractional mean curvature flow of Lipschitz graphs (Annalisa Cesaroni and Matteo Novaga, 2021)4Cesaroni 2021 A distributional approach to nonlocal curvature flows (Filippo Cagnetti, Massimiliano Morini and Dario Reggiani, 2025)2Cagnetti 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Stability results for nonlocal geometric evolutions and limit cases for fractional mean curvature flows

    Annalisa Cesaroni, Lucia De Luca, Matteo Novaga et al.Wider flows

    Topics Anisotropic & nonlocal flows, Weak & singular flows, Other curvature flows · Shared terms “fractional mean”, “regularization”

  2. A distributional approach to nonlocal curvature flows

    Filippo Cagnetti, Massimiliano Morini and Dario ReggianiWider flows

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

  3. Neckpinch singularities in fractional mean curvature flows

    Eleonora Cinti, Carlo Sinestrari and Enrico ValdinociWider flows

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

  4. Fractional mean curvature flow of Lipschitz graphs

    Annalisa Cesaroni and Matteo NovagaWider flows

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

  5. 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”

  6. Convergence of supercritical fractional flows to the mean curvature flow

    Lucia De Luca, Andrea Kubin and Marcello PonsiglioneWider flows

    Topics Anisotropic & nonlocal flows, Weak & singular flows · Shared terms “fractional”

  7. Some results on anisotropic fractional mean curvature flows

    Antonin Chambolle, Matteo Novaga and Berardo RuffiniWider flows

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

  8. Convex sets evolving by volume preserving fractional mean curvature flows

    Eleonora Cinti, Carlo Sinestrari and Enrico ValdinociWider 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) Other curvature flows Wider flows Flows that move a hypersurface or metric by some other function of curvature: Gauss curvature (worn stones), inverse mean curvature (the Penrose… Start with: Shapes of worn stones (Firey, 1974) 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) Weak & singular flows Running a flow when the data isn't smooth or singularities form: Ricci flows from rough initial data, through singularities and on metric measure… Start with: Deformation of C⁰ Riemannian metrics in the direction of their Ricci curvature (Simon, 2002)

In the guidebook

  1. 12C.9
    After PerelmanBook 12C · Extinction and the Proof

    Singular flows, the Smale conjecture, and Bamler’s theory.

  2. 6A.10
    Curve Shortening and the First Geometric FlowsBook 6A · The Heat Equation and Its Relatives

    Shrinking circles, grim reapers, neckpinches and Huisken’s monotonicity, seen in the plane.