Paper · Jun 7, 2018

Implicit time discretization for the mean curvature flow of mean convex sets

Guido De Philippis, Tim Laux

math.APWider flowsarXiv:1806.02716v2
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Abstract

In this note we analyze the Almgren-Taylor-Wang scheme for mean curvature flow in the case of mean convex initial conditions. We show that the scheme preserves strict mean convexity and, by compensated compactness techniques, that the arrival time functions converge strictly in . In particular, this establishes the convergence of the time-integrated perimeters of the approximations. As a corollary, the conditional convergence result of Luckhaus-Sturzenhecker becomes unconditonal in the mean convex case.

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Submitted
Jun 7, 2018
Last revised
May 2, 2019 (v2)
Subjects
math.AP

Comments: 16 pages

Mean curvature flowLimit spaces & RCD

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    Topics Limit spaces & RCD · Same author Tim Laux · Shared terms “time discretization”, “time integrated”, “scheme preserves”

  2. Strong convergence of the thresholding scheme for the mean curvature flow of mean convex sets

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  3. De Giorgi varifold solutions to Mean Curvature Flow: a minimizing movements approach

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

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    Topics Mean curvature flow · Shared terms “convex set”, “mean convex”, “integrated perimeters”

  5. Median filter method for mean curvature flow using a random Jacobi algorithm

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    Topics Mean curvature flow · Same author Tim Laux · Shared terms “discretization”

  6. Convergence of the Allen-Cahn Equation to multi-phase mean-curvature flow

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    Topics Mean curvature flow · Same author Tim Laux · Shared terms “time integrated”, “Sturzenhecker”, “Luckhaus”

  7. Generic level sets in mean curvature flow are BV solutions

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    Topics Mean curvature flow · Same author Tim Laux · Shared terms “Sturzenhecker”, “Luckhaus”, “perimeter”

  8. A mean curvature flow arising in adversarial training

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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) Limit spaces & RCD What manifolds look like in the limit when only a lower Ricci bound is kept. Cheeger–Colding theory describes Ricci limit spaces, and RCD spaces… Start with: On the structure of spaces with Ricci curvature bounded below. I (Cheeger et al., 1997)

In the guidebook

  1. 6C.11
    Optimal Transport and Ricci CurvatureBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Kantorovich duality, Wasserstein geometry, the heat equation as a gradient flow of entropy, displacement convexity, and Ricci curvature as convexity, from Lott–Sturm–Villani to McCann–Topping and the reduced volume.

  2. 9B.4
    Convergence of ManifoldsBook 9B · Comparison, Convergence and Heat on Manifolds

    Gromov–Hausdorff and Cheeger–Gromov convergence, and the compactness theorem.

  3. 11B.3
    Compactness of Ricci FlowsBook 11B · Solitons, Compactness and Singularities

    Hamilton’s compactness theorem and the missing injectivity radius bound.