Paper · Apr 11, 2022

Stability of the ball under volume preserving fractional mean curvature flow

Annalisa Cesaroni, Matteo Novaga

math.APWider flowsarXiv:2204.04923v2
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Abstract

We consider the volume constrained fractional mean curvature flow of a nearly spherical set, and prove long time existence and asymptotic convergence to a ball. The result applies in particular to convex initial data, under the assumption of global existence. Similarly, we show exponential convergence to a constant for the fractional mean curvature flow of a periodic graph.

Mean curvature flowAnisotropic & nonlocal flowsStability & uniquenessLong-time existence & convergence

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 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 Graphical translators for anisotropic and crystalline mean curvature flow (Annalisa Cesaroni, Heiko Kroener and Matteo Novaga, 2021)3Cesaroni 2021 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 Short time existence and smoothness of the nonlocal mean curvature flow of graphs (Anoumou Attiogbe, Mouahmed Moustapha Fall and Tobias Weth, 2022)7Attiogbe 2022 The nonlocal mean curvature flow of periodic graphs (Bogdan-Vasile Matioc and Christoph Walker, 2022)4Matioc 2022 Convergence of the volume preserving fractional mean curvature flow for convex sets (Vesa Julin and Domenico Angelo La Manna, 2023)1Julin 2023 Stability analysis for the anisotropic curve shortening flow of planar networks (Michael Gößwein, Matteo Novaga and Paola Pozzi, 2023)8Gößwein 2023 A distributional approach to nonlocal curvature flows (Filippo Cagnetti, Massimiliano Morini and Dario Reggiani, 2025)5Cagnetti 2025 A semi-discrete scheme for horizontal mean curvature flow in the Heisenberg group (A. Cesaroni, M. Novaga and P. Pozzi, 2026)6Cesaroni 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 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. Graphical translators for anisotropic and crystalline mean curvature flow

    Annalisa Cesaroni, Heiko Kroener and Matteo NovagaWider flows PDF

    Topics Anisotropic & nonlocal flows, Stability & uniqueness · Same authors Annalisa Cesaroni, Matteo Novaga

  4. 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 “periodic graph”

  5. A distributional approach to nonlocal curvature flows

    Filippo Cagnetti, Massimiliano Morini and Dario ReggianiWider flows PDF

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

  6. A semi-discrete scheme for horizontal mean curvature flow in the Heisenberg group

    A. Cesaroni, M. Novaga and P. PozziWider flows PDF

    Topics Mean curvature flow · Same authors Annalisa Cesaroni, Matteo Novaga

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

    Anoumou Attiogbe, Mouahmed Moustapha Fall and Tobias WethWider flows PDF

    Topics Anisotropic & nonlocal flows, Stability & uniqueness · Shared terms “fractional mean”, “time existence”

  8. Stability analysis for the anisotropic curve shortening flow of planar networks

    Michael Gößwein, Matteo Novaga and Paola PozziWider flows PDF

    Topics Anisotropic & nonlocal flows, Stability & uniqueness · Same author Matteo Novaga · Shared terms “initial data”

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. 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) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005) 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.