Paper · Mar 21, 2021

Fractional mean curvature flow of Lipschitz graphs

Annalisa Cesaroni, Matteo Novaga

math.APWider flowsarXiv:2103.11346v3
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Abstract

We consider the fractional mean curvature flow of entire Lipschitz graphs. We provide regularity results, and we study the long time asymptotics of the flow. In particular we show that in a suitable rescaled framework, if the initial graph is a sublinear perturbation of a cone, the evolution asymptotically approaches an expanding self-similar solution. We also prove stability of hyperplanes and of convex cones in the unrescaled setting.

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Submitted
Mar 21, 2021
Last revised
Nov 26, 2021 (v3)
Subjects
math.AP
Mean curvature flowAnisotropic & nonlocal flowsStability & uniqueness

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 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)6Chambolle 2016 Fattening and nonfattening phenomena for planar nonlocal curvature flows (Annalisa Cesaroni, Serena Dipierro, Matteo Novaga et al., 2018)5Cesaroni 2018 Convex sets evolving by volume preserving fractional mean curvature flows (Eleonora Cinti, Carlo Sinestrari and Enrico Valdinoci, 2018)8Cinti 2018 Symmetric self-shrinkers for the fractional mean curvature flow (Annalisa Cesaroni and Matteo Novaga, 2018)3Cesaroni 2018 Stability results for nonlocal geometric evolutions and limit cases for fractional mean curvature flows (Annalisa Cesaroni, Lucia De Luca, Matteo Novaga et al., 2020)4Cesaroni 2020 Anisotropic mean curvature flow of Lipschitz graphs and convergence to self-similar solutions (Annalisa Cesaroni, Heiko Kröner and Matteo Novaga, 2021)1Cesaroni 2021 Graphical translators for anisotropic and crystalline mean curvature flow (Annalisa Cesaroni, Heiko Kroener and Matteo Novaga, 2021)7Cesaroni 2021 Stability of the ball under volume preserving fractional mean curvature flow (Annalisa Cesaroni and Matteo Novaga, 2022)2Cesaroni 2022
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Anisotropic mean curvature flow of Lipschitz graphs and convergence to self-similar solutions

    Annalisa Cesaroni, Heiko Kröner and Matteo NovagaWider flows

    Topics Anisotropic & nonlocal flows, Stability & uniqueness · Same authors Annalisa Cesaroni, Matteo Novaga · Shared terms “Lipschitz graphs”, “entire Lipschitz”, “sublinear perturbation”

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

    Annalisa Cesaroni and Matteo NovagaWider flows

    Topics Anisotropic & nonlocal flows, Stability & uniqueness · Same authors Annalisa Cesaroni, Matteo Novaga · Shared terms “fractional mean”, “long time”

  3. Symmetric self-shrinkers for the fractional mean curvature flow

    Annalisa Cesaroni and Matteo NovagaWider flows

    Topics Anisotropic & nonlocal flows, Stability & uniqueness · Same authors Annalisa Cesaroni, Matteo Novaga · Shared terms “fractional mean”

  4. Stability results for nonlocal geometric evolutions and limit cases for fractional mean curvature flows

    Annalisa Cesaroni, Lucia De Luca, Matteo Novaga et al.Wider flows

    Topics Anisotropic & nonlocal flows, Stability & uniqueness · Same authors Annalisa Cesaroni, Matteo Novaga · Shared terms “fractional mean”

  5. Fattening and nonfattening phenomena for planar nonlocal curvature flows

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

    Topics Anisotropic & nonlocal flows, Stability & uniqueness · Same authors Annalisa Cesaroni, Matteo Novaga · Shared terms “fractional mean”, “Lipschitz”

  6. Some results on anisotropic fractional mean curvature flows

    Antonin Chambolle, Matteo Novaga and Berardo RuffiniWider flows

    Topics Anisotropic & nonlocal flows · Same author Matteo Novaga · Shared terms “fractional mean”

  7. Graphical translators for anisotropic and crystalline mean curvature flow

    Annalisa Cesaroni, Heiko Kroener and Matteo NovagaWider flows

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

  8. Convex sets evolving by volume preserving fractional mean curvature flows

    Eleonora Cinti, Carlo Sinestrari and Enrico ValdinociWider flows

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

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) 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)

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.