Paper · Jun 26, 2019

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

Vesa Julin, Domenico La Manna

math.APWider flowsarXiv:1906.10990v2
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Abstract

We establish short-time existence of the smooth solution to the fractional mean curvature flow when the initial set is bounded and C^1,1-regular. We provide the same result also for the volume preserving fractional mean curvature flow.

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Submitted
Jun 26, 2019
Last revised
Apr 22, 2020 (v2)
Subjects
math.AP, math.DG
Mean curvature flowAnisotropic & nonlocal flowsHarmonic maps & DeTurck trick

Its neighbourhood

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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)7Chambolle 2016 Neckpinch singularities in fractional mean curvature flows (Eleonora Cinti, Carlo Sinestrari and Enrico Valdinoci, 2016)6Cinti 2016 Convex sets evolving by volume preserving fractional mean curvature flows (Eleonora Cinti, Carlo Sinestrari and Enrico Valdinoci, 2018)4Cinti 2018 Stability of the ball under volume preserving fractional mean curvature flow (Annalisa Cesaroni and Matteo Novaga, 2022)2Cesaroni 2022 Asymptotic of the Discrete Volume-Preserving Fractional Mean Curvature Flow via a Nonlocal Quantitative Alexandrov Theorem (De Gennaro Daniele, Andrea Kubin and Anna Kubin, 2022)8Daniele 2022 Short time existence and smoothness of the nonlocal mean curvature flow of graphs (Anoumou Attiogbe, Mouahmed Moustapha Fall and Tobias Weth, 2022)3Attiogbe 2022 Convergence of the volume preserving fractional mean curvature flow for convex sets (Vesa Julin and Domenico Angelo La Manna, 2023)1Julin 2023 Existence of the classical solution to the fractional mean curvature flow with capillary-type boundary conditions (Linlin Fan and Peibiao Zhao, 2026)5Fan 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Convergence of the volume preserving fractional mean curvature flow for convex sets

    Vesa Julin and Domenico Angelo La MannaWider flows

    Topics Anisotropic & nonlocal flows · Same authors Vesa Julin, Domenico La Manna · Shared terms “fractional mean”, “preserving fractional”, “volume preserving”

  2. 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”, “preserving fractional”, “volume preserving”

  3. Short time existence and smoothness of the nonlocal mean curvature flow of graphs

    Anoumou Attiogbe, Mouahmed Moustapha Fall and Tobias WethWider flows

    Topics Anisotropic & nonlocal flows, Harmonic maps & DeTurck trick · Shared terms “short time”, “fractional mean”, “classical solution”

  4. 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”, “preserving fractional”, “volume preserving”

  5. Existence of the classical solution to the fractional mean curvature flow with capillary-type boundary conditions

    Linlin Fan and Peibiao ZhaoWider flows

    Topics Anisotropic & nonlocal flows, Harmonic maps & DeTurck trick · Shared terms “fractional mean”, “classical solution”, “short time”

  6. Neckpinch singularities in fractional mean curvature flows

    Eleonora Cinti, Carlo Sinestrari and Enrico ValdinociWider flows

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

  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. Asymptotic of the Discrete Volume-Preserving Fractional Mean Curvature Flow via a Nonlocal Quantitative Alexandrov Theorem

    De Gennaro Daniele, Andrea Kubin and Anna KubinWider flows

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

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) Harmonic maps & DeTurck trick Ricci flow is only weakly parabolic, because it is invariant under diffeomorphisms. The DeTurck trick adds a Lie-derivative term that makes it… Start with: Harmonic Mappings of Riemannian Manifolds (Eells et al., 1964)

In the guidebook

  1. 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.