Paper · Nov 21, 2018

Convex sets evolving by volume preserving fractional mean curvature flows

Eleonora Cinti, Carlo Sinestrari, Enrico Valdinoci

math.APWider flowsarXiv:1811.08651
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Abstract

We consider the volume preserving geometric evolution of the boundary of a set under fractional mean curvature. We show that smooth convex solutions maintain their fractional curvatures bounded for all times, and the long time asymptotics approach round spheres. The proofs are based on apriori estimates on the inner and outer radii of the solutions.

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Submitted
Nov 21, 2018
Subjects
math.AP
Journal
Analysis & PDE 13 (2020) 2149-2171
DOI
10.2140/apde.2020.13.2149
Mean curvature flowAnisotropic & nonlocal flows

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)5Chambolle 2016 Neckpinch singularities in fractional mean curvature flows (Eleonora Cinti, Carlo Sinestrari and Enrico Valdinoci, 2016)1Cinti 2016 Fattening and nonfattening phenomena for planar nonlocal curvature flows (Annalisa Cesaroni, Serena Dipierro, Matteo Novaga et al., 2018)8Cesaroni 2018 Short time existence of the classical solution to the fractional Mean curvature flow (Vesa Julin and Domenico La Manna, 2019)4Julin 2019 Fractional mean curvature flow of Lipschitz graphs (Annalisa Cesaroni and Matteo Novaga, 2021)7Cesaroni 2021 Stability of the ball under volume preserving fractional mean curvature flow (Annalisa Cesaroni and Matteo Novaga, 2022)3Cesaroni 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)6Daniele 2022 Convergence of the volume preserving fractional mean curvature flow for convex sets (Vesa Julin and Domenico Angelo La Manna, 2023)2Julin 2023
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Neckpinch singularities in fractional mean curvature flows

    Eleonora Cinti, Carlo Sinestrari and Enrico ValdinociWider flows

    Topics Anisotropic & nonlocal flows · Same authors Eleonora Cinti, Carlo Sinestrari, Enrico Valdinoci · Shared terms “fractional mean”

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

    Vesa Julin and Domenico Angelo La MannaWider flows

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

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

    Annalisa Cesaroni and Matteo NovagaWider flows

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

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

    Vesa Julin and Domenico La MannaWider flows

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

  5. Some results on anisotropic fractional mean curvature flows

    Antonin Chambolle, Matteo Novaga and Berardo RuffiniWider flows

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

  6. 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 “preserving fractional”, “fractional mean”, “volume preserving”

  7. Fractional mean curvature flow of Lipschitz graphs

    Annalisa Cesaroni and Matteo NovagaWider flows

    Topics Anisotropic & nonlocal flows · Shared terms “fractional mean”, “time asymptotics”, “long time”

  8. Fattening and nonfattening phenomena for planar nonlocal curvature flows

    Annalisa Cesaroni, Serena Dipierro, Matteo Novaga et al.Wider flows

    Topics Anisotropic & nonlocal flows · Same author Enrico Valdinoci · Shared terms “fractional mean”, “geometric evolution”

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)

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.