Paper · May 13, 2021

Anisotropic mean curvature flow of Lipschitz graphs and convergence to self-similar solutions

Annalisa Cesaroni, Heiko Kröner, Matteo Novaga

math.APWider flowsarXiv:2105.06359
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Abstract

We consider the anisotropic mean curvature flow of entire Lipschitz graphs. We prove existence and uniqueness of expanding self-similar solutions which are asymptotic to a prescribed cone, and we characterize the long time behavior of solutions, after suitable rescaling, when the initial datum is a sublinear perturbation of a cone. In the case of regular anisotropies, we prove the stability of self-similar solutions asymptotic to strictly mean convex cones, with respect to perturbations vanishing at infinity. We also show the stability of hyperplanes, with a proof which is novel also for the isotropic mean curvature flow.

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Submitted
May 13, 2021
Subjects
math.AP
Mean curvature flowAnisotropic & nonlocal flowsStability & uniquenessLong-time existence & convergence

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 Existence and uniqueness for anisotropic and crystalline mean curvature flows (Antonin Chambolle, Massimiliano Morini, Matteo Novaga et al., 2017)8Chambolle 2017 Fattening and nonfattening phenomena for planar nonlocal curvature flows (Annalisa Cesaroni, Serena Dipierro, Matteo Novaga et al., 2018)4Cesaroni 2018 Symmetric self-shrinkers for the fractional mean curvature flow (Annalisa Cesaroni and Matteo Novaga, 2018)5Cesaroni 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)6Cesaroni 2020 Anisotropic and crystalline mean curvature flow of mean-convex sets (Antonin Chambolle and Matteo Novaga, 2020)7Chambolle 2020 Fractional mean curvature flow of Lipschitz graphs (Annalisa Cesaroni and Matteo Novaga, 2021)1Cesaroni 2021 Graphical translators for anisotropic and crystalline mean curvature flow (Annalisa Cesaroni, Heiko Kroener and Matteo Novaga, 2021)3Cesaroni 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. Fractional mean curvature flow of Lipschitz graphs

    Annalisa Cesaroni 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, Long-time existence & convergence, Stability & uniqueness · Same authors Annalisa Cesaroni, Matteo Novaga

  3. 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 · Shared terms “anisotropy”

  4. 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 “initial datum”, “Lipschitz”

  5. 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

  6. 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 “characterize”

  7. Anisotropic and crystalline mean curvature flow of mean-convex sets

    Antonin Chambolle and Matteo NovagaWider flows

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

  8. Existence and uniqueness for anisotropic and crystalline mean curvature flows

    Antonin Chambolle, Massimiliano Morini, Matteo Novaga et al.Wider flows

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

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) 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.