Paper · Aug 11, 2017

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

Zhou Gang

math.APWider flowsarXiv:1708.03484v2
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Abstract

We study the formation of generic singularities of mean curvature flow by combining the different approaches, specifically the methods in studying blowup of nonlinear heat equations, the techniques used by the author and the collaborators for mean curvature flow, and these invented by Colding and Minicozzi. We study the solution in a neighborhood of the blowup point, in some generic regimes, we find the key parameters take favorable signs and have sharp decay rates. We provide the remainder estimates in different norms.

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Submitted
Aug 11, 2017
Last revised
Jul 26, 2021 (v2)
Subjects
math.AP, math.DG

Comments: 71 pages

Mean curvature flowSingularities

Its neighbourhood

Connected papers

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.

Connected papers This paper in the centre and the 8 papers closest to it around it, in date order clockwise from the top. The closer a paper sits to the centre, the stronger the connection. This paper Entropy and generic mean curvature flow in curved ambient spaces (Alexander Mramor, 2017)6Mramor 2017 On the mean convexity of a space-and-time neighborhood of generic singularities formed by mean curvature flow (Zhou Gang, 2018)1Gang 2018 Local Entropy and Generic Multiplicity One Singularities of Mean Curvature Flow of Surfaces (Ao Sun, 2018)4Sun 2018 Precise asymptotics near a generic S¹×ℝ³ singularity of mean curvature flow (Zhou Gang and Shengwen Wang, 2019)2Gang 2019 Entropy and codimension bounds for generic singularities (Tobias Holck Colding and William P. Minicozzi, 2019)3Colding 2019 On the non-degenerate and degenerate generic singularities formed by mean curvature flow (Gang Zhou, 2021)5Zhou 2021 Approximation of mean curvature flow with generic singularities by smooth flows with surgery (Joshua Daniels-Holgate, 2021)7Daniels-Holg 2021 Mean curvature flow with generic low-entropy initial data II (Otis Chodosh, Christos Mantoulidis and Felix Schulze, 2023)8Chodosh 2023
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. On the mean convexity of a space-and-time neighborhood of generic singularities formed by mean curvature flow

    Zhou GangWider flows

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

  2. 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”

  3. Entropy and codimension bounds for generic singularities

    Tobias Holck Colding and William P. MinicozziWider flows

    Topics Singularities · Shared terms “generic singularities”

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

    Ao SunWider flows

    Topics Singularities · Shared terms “generic singularities”, “Minicozzi”, “Colding”

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

    Gang ZhouWider flows

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

  6. Entropy and generic mean curvature flow in curved ambient spaces

    Alexander MramorWider flows

    Topics Singularities · Shared terms “generic singularities”, “Minicozzi”, “Colding”

  7. Approximation of mean curvature flow with generic singularities by smooth flows with surgery

    Joshua Daniels-HolgateWider flows

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

  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

  1. 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.

  2. 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.

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

    Limits of rescaled manifolds and flows: why blow-up limits are nonnegatively curved, what blow-downs see, and how lines appear at infinity.