Paper · Apr 15, 2022

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

De Gennaro Daniele, Andrea Kubin, Anna Kubin

math.APWider flowsarXiv:2204.07450v3
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Abstract

We characterize the long time behaviour of a discrete-in-time approximation of the volume preserving fractional mean curvature flow. In particular, we prove that the discrete flow starting from any bounded set of finite fractional perimeter converges exponentially fast to a single ball. As an intermediate result we establish a quantitative Alexandrov type estimate in the fractional setting for normal deformations of a ball. Finally, we provide existence for flat flows as limit points of the discrete flow when the time discretization parameter tends to zero.

Comments: Added acknowledgements, corrected some typos

Mean curvature flowAnisotropic & nonlocal flowsLong-time existence & convergence

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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 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 Long Time Behaviour of the Discrete Volume Preserving Mean Curvature Flow in the Flat Torus (Daniele De Gennaro and Anna Kubin, 2022)3De Gennaro 2022 Stability of the ball under volume preserving fractional mean curvature flow (Annalisa Cesaroni and Matteo Novaga, 2022)1Cesaroni 2022 The nonlocal mean curvature flow of periodic graphs (Bogdan-Vasile Matioc and Christoph Walker, 2022)6Matioc 2022 Stability of the surface diffusion flow and volume-preserving mean curvature flow in the flat torus (Daniele De Gennaro, Antonia Diana, Andrea Kubin et al., 2023)4De Gennaro 2023 Convergence of the volume preserving fractional mean curvature flow for convex sets (Vesa Julin and Domenico Angelo La Manna, 2023)2Julin 2023 The asymptotic of the Mullins-Sekerka and the area-preserving curvature flow in the planar flat torus (Vedansh Arya, Daniele De Gennaro and Anna Kubin, 2025)8Arya 2025 A distributional approach to nonlocal curvature flows (Filippo Cagnetti, Massimiliano Morini and Dario Reggiani, 2025)7Cagnetti 2025 A variational approach to the volume-preserving anisotropic mean curvature flow in 2D (Andrea Kubin, Domenico Angelo La Manna and Enrico Pasqualetto, 2025)5Kubin 2025
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. Convergence of the volume preserving fractional mean curvature flow for convex sets

    Vesa Julin and Domenico Angelo La MannaWider flows PDF

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

  3. Long Time Behaviour of the Discrete Volume Preserving Mean Curvature Flow in the Flat Torus

    Daniele De Gennaro and Anna KubinWider flows PDF

    Topics Long-time existence & convergence · Same author Anna Kubin · Shared terms “discrete volume”, “time behaviour”, “volume preserving”

  4. Stability of the surface diffusion flow and volume-preserving mean curvature flow in the flat torus

    Daniele De Gennaro, Antonia Diana, Andrea Kubin et al.Wider flows PDF

    Topics Long-time existence & convergence · Same authors Andrea Kubin, Anna Kubin · Shared terms “volume preserving”, “exponentially fast”, “perimeter”

  5. 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 Andrea Kubin · Shared terms “volume preserving”

  6. 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”, “converges exponentially”

  7. A distributional approach to nonlocal curvature flows

    Filippo Cagnetti, Massimiliano Morini and Dario ReggianiWider flows PDF

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

  8. The asymptotic of the Mullins-Sekerka and the area-preserving curvature flow in the planar flat torus

    Vedansh Arya, Daniele De Gennaro and Anna KubinWider flows PDF

    Topics Long-time existence & convergence · Same author Anna Kubin · Shared terms “quantitative Alexandrov”, “flat flow”, “exponentially fast”

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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) 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)

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.