Paper · Jul 8, 2023

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

Vesa Julin, Domenico Angelo La Manna

math.APWider flowsarXiv:2307.03912v3
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Abstract

We prove that the volume preserving fractional mean curvature flow starting from a convex set does not develop singularities along the flow. By the recent result of Cesaroni-Novaga this then implies that the flow converges to a ball exponentially fast. In the proof we show that the apriori estimates due to Cinti-Sinestrari-Valdinoci imply the -regularity of the flow and then provide a regularity argument which improves this into -regularity of the flow. The regularity step from into does not rely on convexity and can probably be adopted to more general setting.

Mean curvature flowAnisotropic & nonlocal flowsSingularitiesLong-time existence & convergence

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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 2021, 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 Stability of the ball under volume preserving fractional mean curvature flow (Annalisa Cesaroni and Matteo Novaga, 2022)1Cesaroni 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)2Daniele 2022 Consistency of the flat flow solution to the volume preserving mean curvature flow (Vesa Julin and Joonas Niinikoski, 2022)7Julin 2022 The nonlocal mean curvature flow of periodic graphs (Bogdan-Vasile Matioc and Christoph Walker, 2022)5Matioc 2022 A distributional approach to nonlocal curvature flows (Filippo Cagnetti, Massimiliano Morini and Dario Reggiani, 2025)6Cagnetti 2025 A variational approach to the volume-preserving anisotropic mean curvature flow in 2D (Andrea Kubin, Domenico Angelo La Manna and Enrico Pasqualetto, 2025)3Kubin 2025 Existence of the classical solution to the fractional mean curvature flow with capillary-type boundary conditions (Linlin Fan and Peibiao Zhao, 2026)8Fan 2026 Eventual regularity of the volume-preserving mean curvature flow in three and two dimensions (Vedansh Arya, Seongmin Jeon and Vesa Julin, 2026)4Arya 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Stability of the ball under volume preserving fractional mean curvature flow

    Annalisa Cesaroni and Matteo NovagaWider flows PDF

    Topics Anisotropic & nonlocal flows, Long-time existence & convergence · Shared terms “preserving fractional”, “fractional mean”, “volume preserving”

  2. 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 PDF

    Topics Anisotropic & nonlocal flows, Long-time existence & convergence · Shared terms “preserving fractional”, “fractional mean”, “volume preserving”

  3. A variational approach to the volume-preserving anisotropic mean curvature flow in 2D

    Andrea Kubin, Domenico Angelo La Manna and Enrico PasqualettoWider flows PDF

    Topics Anisotropic & nonlocal flows, Long-time existence & convergence · Same author Domenico Angelo La Manna · Shared terms “volume preserving”

  4. Eventual regularity of the volume-preserving mean curvature flow in three and two dimensions

    Vedansh Arya, Seongmin Jeon and Vesa JulinWider flows PDF

    Topics Long-time existence & convergence, Singularities · Same author Vesa Julin · Shared terms “volume preserving”, “exponentially fast”

  5. The nonlocal mean curvature flow of periodic graphs

    Bogdan-Vasile Matioc and Christoph WalkerWider flows PDF

    Topics Anisotropic & nonlocal flows, Long-time existence & convergence · Shared terms “exponentially fast”

  6. A distributional approach to nonlocal curvature flows

    Filippo Cagnetti, Massimiliano Morini and Dario ReggianiWider flows PDF

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

  7. Consistency of the flat flow solution to the volume preserving mean curvature flow

    Vesa Julin and Joonas NiinikoskiWider flows PDF

    Topics Mean curvature flow · Same author Vesa Julin · Shared terms “volume preserving”, “flow starting”

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

    Linlin Fan and Peibiao ZhaoWider flows PDF

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

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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. 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 usually cannot run forever: curvature blows up in finite time, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993) Long-time existence & convergence A geometric flow either runs forever or stops because something blows up. Extension criteria say which quantity must blow up at a finite singular… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982)

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