Paper · Apr 8, 2025

A distributional approach to nonlocal curvature flows

Filippo Cagnetti, Massimiliano Morini, Dario Reggiani

math.APWider flowsarXiv:2504.06162v2
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Abstract

In a novel distributional approach has been introduced to provide a well-posed formulation of a class of crystalline mean curvature flows. In this paper, such an approach is extended to the nonlocal setting. Applications include the fractional mean curvature flow and the Minkowski flow; i.e., the geometric flow generated by the -dimensional Minkowski pre-content.

Mean curvature flowOther curvature flowsAnisotropic & nonlocal flows

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)4Cesaroni 2021 Stability of the ball under volume preserving fractional mean curvature flow (Annalisa Cesaroni and Matteo Novaga, 2022)6Cesaroni 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)3Daniele 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 Convergence of the volume preserving fractional mean curvature flow for convex sets (Vesa Julin and Domenico Angelo La Manna, 2023)8Julin 2023 Discrete-to-continuum crystalline curvature flows (Antonin Chambolle, Daniele De Gennaro and Massimiliano Morini, 2024)1Chambolle 2024 Deforming Locally Convex Curves into Curves of Constant k-order Width (Laiyuan Gao, Horst Martini and Deyan Zhang, 2024)5Gao 2024 Variational Nonlinear and Nonlocal Curvature Flows (Daniele De Gennaro, 2025)2De Gennaro 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Discrete-to-continuum crystalline curvature flows

    Antonin Chambolle, Daniele De Gennaro and Massimiliano MoriniWider flows PDF

    Topics Anisotropic & nonlocal flows, Other curvature flows · Same author Massimiliano Morini · Shared terms “crystalline”

  2. Variational Nonlinear and Nonlocal Curvature Flows

    Daniele De GennaroWider flows PDF

    Topics Anisotropic & nonlocal flows, Other curvature flows

  3. 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 · Shared terms “fractional mean”

  4. Graphical translators for anisotropic and crystalline mean curvature flow

    Annalisa Cesaroni, Heiko Kroener and Matteo NovagaWider flows PDF

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

  5. Deforming Locally Convex Curves into Curves of Constant -order Width

    Laiyuan Gao, Horst Martini and Deyan ZhangWider flows PDF

    Topics Anisotropic & nonlocal flows, Other curvature flows

  6. Stability of the ball under volume preserving fractional mean curvature flow

    Annalisa Cesaroni and Matteo NovagaWider flows PDF

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

  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 · Shared terms “fractional mean”

  8. 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 · 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. 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)

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.