Paper · Mar 29, 2018

On the mean convexity of a space-and-time neighborhood of generic singularities formed by mean curvature flow

Zhou Gang

math.APWider flowsarXiv:1803.10903v3
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Abstract

We consider one of the generic regimes of formation of singularities. We obtain a detailed description of a possibly small, but fixed, neighborhood of the blowup point, up to (and including) the blowup time, and find that it is mean convex. This confirms a conjecture by Ilmanen. And we find that the singularity is isolated from the other ones.

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Submitted
Mar 29, 2018
Last revised
May 22, 2019 (v3)
Subjects
math.AP, math.DG

Comments: 72 Pages

Mean curvature flowSingularities

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.

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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. On the dynamics of formation of generic singularities of mean curvature flow

    Zhou GangWider flows

    Topics Singularities · Same author Zhou Gang · Shared terms “generic singularities”, “blowup point”, “generic regimes”

  2. On the non-degenerate and degenerate generic singularities formed by mean curvature flow

    Gang ZhouWider flows

    Topics Singularities · Shared terms “singularities formed”, “generic singularities”, “neighborhood”

  3. Precise asymptotics near a generic singularity of mean curvature flow

    Zhou Gang and Shengwen WangWider flows

    Topics Singularities · Same author Zhou Gang · Shared terms “generic singularities”, “neighborhood”

  4. Entropy and codimension bounds for generic singularities

    Tobias Holck Colding and William P. MinicozziWider flows

    Topics Singularities · Shared terms “generic singularities”

  5. Local Entropy and Generic Multiplicity One Singularities of Mean Curvature Flow of Surfaces

    Ao SunWider flows

    Topics Singularities · Shared terms “generic singularities”, “Ilmanen”

  6. Ancient low entropy flows, mean convex neighborhoods, and uniqueness

    Kyeongsu Choi, Robert Haslhofer and Or HershkovitsWider flows

    Topics Singularities · Shared terms “time neighborhood”, “mean convex”, “Ilmanen”

  7. On the rigidity of mean convex self-shrinkers

    Qiang Guang and Jonathan J. ZhuWider flows

    Topics Singularities · Shared terms “detailed description”, “mean convex”, “Ilmanen”

  8. Mean curvature flow with generic low-entropy initial data II

    Otis Chodosh, Christos Mantoulidis and Felix SchulzeWider flows

    Topics Singularities · Shared terms “generic singularities”

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) Singularities On a closed manifold the flow can stop in finite time: curvature blows up, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993)

In the guidebook

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    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.

  3. 11B.4
    Blow-Ups, Blow-Downs and Splitting at InfinityBook 11B · Solitons, Compactness and Singularities

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