Paper · Dec 4, 2024

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

Alexander Mramor, Ao Sun

math.DGWider flowsarXiv:2412.03475v2
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Abstract

In this article we show that generally almost regular flows, introduced by Bamler and Kleiner, in closed 3-manifolds will either go extinct in finite time or flow to a collection of smooth embedded minimal surfaces, possibly with multiplicity. Using a perturbative argument then we construct piecewise almost regular flows which either go extinct in finite time or flow to a stable minimal surface, possibly with multiplicity. We apply these results to construct minimal surfaces in 3-manifolds in a variety of circumstances, mainly novel from the point of the view that the arguments are via parabolic methods.

Comments: Some minor edits made, to appear in Journal of the London Mathematical Society

Mean curvature flowSingularities3-manifoldsMinimal & CMC surfaces

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    Topics Mean curvature flow · Same author Ao Sun · Shared terms “multiplicity”, “smooth embedded”

  3. The Dirichlet problem for the minimal surface system on smooth domains

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    Topics Minimal & CMC surfaces · Shared terms “long time”

  4. Existence of 5 minimal tori in 3-spheres of positive Ricci curvature

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    Topics Minimal & CMC surfaces · Shared terms “embedded minimal”

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

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    Topics Minimal & CMC surfaces, Singularities

  6. Minimal hypersurfaces in by doubling the equatorial

    Nikolaos Kapouleas and Jiahua ZouWider flows PDF

    Topics Minimal & CMC surfaces, 3-manifolds

  7. Classification of Minimal Surfaces and Solitons to the Mean Curvature Flow in as Translation Surfaces

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    Topics Minimal & CMC surfaces

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

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    Topics Minimal & CMC surfaces, Singularities · Shared terms “possibly”

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) 3-manifolds Dimension three is where Ricci flow has done its most complete work. It proves geometrization, describes the flow's long-time behaviour, and now… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) 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. 10A.1
    A Zoo of Three-ManifoldsBook 10A · Three-Manifolds and Geometrization

    The 3-sphere, lens spaces, the Poincaré sphere, the Hopf fibration and the shape of space.

  2. 10A.2
    Finite Rotation Groups and Spherical Space FormsBook 10A · Three-Manifolds and Geometrization

    Klein's classification by counting poles, the binary groups, and every quotient of the round 3-sphere.

  3. 10A.4
    Homotopy Groups, Degree and the Hurewicz TheoremBook 10A · Three-Manifolds and Geometrization

    Spheres mapped into manifolds: the algebraic topology that finite extinction runs on.