Paper · Nov 27, 2023

On the Structure of Singularities of Weak Mean Curvature Flows with Mean Curvature Bounds

Maxwell Stolarski

math.DGWider flowsarXiv:2311.16262
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Abstract

This paper studies singularities of mean curvature flows with integral mean curvature bounds for some . For such flows, any tangent flow is given by the flow of a stationary cone . When and is a regular cone, we prove that the tangent flow is unique. These results hold for general integral Brakke flows of arbitrary codimension in an open subset with . For smooth, codimension one mean curvature flows with , we also show that, at points where a tangent flow is given by an area-minimizing Simons cone, there is an accompanying limit flow given by a smooth Hardt-Simon minimal surface.

Comments: 33 pages, comments welcome

Mean curvature flowSingularitiesWeak & singular flowsMinimal & CMC surfaces

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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
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    Junyoung ParkWider flows PDF

    Topics Minimal & CMC surfaces · Shared terms “Hardt Simon”, “stationary”

  2. New advances on the existence and regularity of Brakke's mean curvature flows

    Salvatore StuvardWider flows PDF

    Topics Minimal & CMC surfaces, Weak & singular flows, Singularities · Shared terms “Brakke flows”

  3. Mean curvature flow converging to an minimizing cone and its Hardt-Simon foliation

    Jiuzhou HuangWider flows PDF

    Topics Singularities · Shared terms “Hardt Simon”, “area minimizing”

  4. On the long-time limit of the mean curvature flow in closed manifolds

    Alexander Mramor and Ao SunWider flows PDF

    Topics Minimal & CMC surfaces, Singularities

  5. When can minimal hypersurfaces be connected by mean curvature flow?

    Jingwen Chen and Pedro GasparWider flows PDF

    Topics Minimal & CMC surfaces

  6. The Parabolic -Higgs Equations and Codimension-two Mean Curvature Flows

    Davide Parise, Alessandro Pigati and Daniel SternWider flows PDF

    Topics Weak & singular flows · Shared terms “Brakke flows”, “integral Brakke”, “stationary”

  7. Uniqueness results and enclosure properties for hypersurfaces with boundary in weighted cylinders

    Katherine Castro and César RosalesWider flows PDF

    Topics Minimal & CMC surfaces, Singularities

  8. Minimal hypersurfaces in by doubling the equatorial

    Nikolaos Kapouleas and Jiahua ZouWider flows PDF

    Topics Minimal & CMC surfaces, Weak & singular flows

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. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) Singularities On a closed manifold the flow usually cannot run forever: curvature blows up in finite time, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993) 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) Minimal & CMC surfaces Minimal surfaces are critical points of area, with zero mean curvature; constant mean curvature (CMC) surfaces are critical for area at fixed volume. Start with: On the structure of manifolds with positive scalar curvature (Schoen et al., 1979)

In the guidebook

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

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

  2. 6A.13
    Self-Similar Variables and MonotonicityBook 6A · The Heat Equation and Its Relatives

    Changing variables so that self-similar solutions stand still: convergence to the Gaussian, Giga–Kohn blow-up, Gaussian-weighted monotonicity formulas, frequency functions, and the blow-up logic of Huisken, Struwe and Perelman.

  3. 6C.8
    Monotonicity, ε-Regularity and Tangent ConesBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Monotonicity formulas for minimal surfaces and harmonic maps, small energy implies smoothness, bubble trees, tangent cones and dimension reduction, with soap films, liquid crystals and pseudolocality.