Paper · Mar 4, 2020

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

Annalisa Cesaroni, Lucia De Luca, Matteo Novaga, Marcello Ponsiglione

math.APWider flowsarXiv:2003.02248
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Abstract

We introduce a notion of uniform convergence for local and nonlocal curvatures. Then, we propose an abstract method to prove the convergence of the corresponding geometric flows, within the level set formulation. We apply such a general theory to characterize the limits of -fractional mean curvature flows as and . In analogy with the -fractional mean curvature flows, we introduce the notion of -Riesz curvature flows and characterize its limit as . Eventually, we discuss the limit behavior as of the flow generated by a regularization of the -Minkowski content.

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Submitted
Mar 4, 2020
Subjects
math.AP

Comments: 26 pages

Mean curvature flowOther curvature flowsAnisotropic & nonlocal flowsWeak & singular flowsStability & uniqueness

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Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Convergence of supercritical fractional flows to the mean curvature flow

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    Topics Anisotropic & nonlocal flows, Weak & singular flows · Same authors Lucia De Luca, Marcello Ponsiglione · Shared terms “nonlocal geometric”, “fractional”, “geometric evolution”

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

    Annalisa Cesaroni and Matteo NovagaWider flows

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  3. 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 “fractional mean”

  4. Fattening and nonfattening phenomena for planar nonlocal curvature flows

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

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

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

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    Topics Anisotropic & nonlocal flows, Other curvature flows · Shared terms “fractional mean”, “flow generated”, “geometric flows”

  8. Eventual Regularization of Fractional Mean Curvature Flow

    Stephen CameronWider flows

    Topics Anisotropic & nonlocal flows, Weak & singular flows, Other curvature flows · Shared terms “fractional mean”, “regularization”

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) 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) Weak & singular flows Running a flow when the data isn't smooth or singularities form: Ricci flows from rough initial data, through singularities and on metric measure… Start with: Deformation of C⁰ Riemannian metrics in the direction of their Ricci curvature (Simon, 2002)

In the guidebook

  1. 12C.9
    After PerelmanBook 12C · Extinction and the Proof

    Singular flows, the Smale conjecture, and Bamler’s theory.

  2. 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.