Paper · Feb 16, 2021

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

Gang Zhou

math.DGWider flowsarXiv:2102.08353v2
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Abstract

In this paper we study a neighborhood of generic singularities formed by mean curvature flow (MCF). We limit our consideration to the singularities modelled on because, compared to the cases with , the present case has the fewest possibilities to be considered. For various possibilities, we provide a detailed description for a small, but fixed, neighborhood of singularity, and prove that a small neighborhood of the singularity is mean convex, and the singularity is isolated. For the remaining possibilities, we conjecture that an entire neighborhood of the singularity becomes singular at the time of blowup, and present evidences to support this conjecture. A key technique is that, when looking for a dominating direction for the rescaled MCF, we need a normal form transformation, as a result, the rescaled MCF is parametrized over some chosen curved cylinder, instead over a standard straight one. This is a long paper. The introduction is carefully written to present the key steps and ideas.

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Submitted
Feb 16, 2021
Last revised
Jul 26, 2021 (v2)
Subjects
math.DG, math.AP

Comments: 92 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 mean convexity of a space-and-time neighborhood of generic singularities formed by mean curvature flow

    Zhou GangWider flows

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

  2. Generic mean curvature flows with cylindrical singularities I: the normal forms and nondegeneracy

    Ao Sun and Jinxin XueWider flows

    Topics Singularities · Shared terms “normal form”, “neighborhood”, “rescaled”

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

    Zhou GangWider flows

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

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

    Zhou Gang and Shengwen WangWider flows

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

  5. On the Stability of Cylindrical Singularities of the Mean Curvature Flow

    Jingxuan ZhangWider flows

    Topics Singularities · Shared terms “rescaled MCF”

  6. Entropy and codimension bounds for generic singularities

    Tobias Holck Colding and William P. MinicozziWider flows

    Topics Singularities · Shared terms “generic singularities”

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

    Ao SunWider flows

    Topics Singularities · Shared terms “generic singularities”

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

    Alexander MramorWider 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)

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