Paper · Aug 2, 2021

A local estimate for the mean curvature flow

Zhen Wang

math.DGWider flowsarXiv:2108.00746
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Abstract

We establish a pointwise estimate of A along the mean curvature flow in terms of the initial geometry and the jHAj bound. As corollaries we obtain the extension theorem of HA and the blowup rate estimate of HA.

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Submitted
Aug 2, 2021
Subjects
math.DG

Comments: 16 pages

Mean curvature flowSingularitiesLong-time existence & convergence

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 2021, 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 On the Stability of Cylindrical Singularities of the Mean Curvature Flow (Jingxuan Zhang, 2021)4Zhang 2021 Singularity formation along the line bundle mean curvature flow (Yu Hin Chan and Adam Jacob, 2023)7Chan 2023 Mean Curvature Flow of High Codimension in Complex Projective Space (Artemis A. Vogiatzi, 2023)8Vogiatzi 2023 Ancient mean curvature flows with finite total curvature (Kyeongsu Choi, Jiuzhou Huang and Taehun Lee, 2024)3Choi 2024 Quantitative uniqueness for mean curvature flow (Tobias Holck Colding and William P. Minicozzi, 2025)6Colding 2025 On the rate of convergence of cylindrical singularity in mean curvature flow (Yiqi Huang and Xinrui Zhao, 2025)2Huang 2025 Solutions to Mean Curvature Flow with Uniform Bounds on the Mean Curvature and Its Gradient (Priyamvada Vishwamitra, 2026)5Vishwamitra 2026 Singularities of Mean Curvature Flow with Bounded Total Curvature (Yongheng Han and Zhen Wang, 2026)1Han 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Singularities of Mean Curvature Flow with Bounded Total Curvature

    Yongheng Han and Zhen WangWider flows

    Topics Singularities · Same author Zhen Wang

  2. On the rate of convergence of cylindrical singularity in mean curvature flow

    Yiqi Huang and Xinrui ZhaoWider flows

    Topics Long-time existence & convergence, Singularities

  3. Ancient mean curvature flows with finite total curvature

    Kyeongsu Choi, Jiuzhou Huang and Taehun LeeWider flows

    Topics Long-time existence & convergence · Shared terms “pointwise estimate”, “corollary”

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

    Jingxuan ZhangWider flows

    Topics Long-time existence & convergence, Singularities

  5. Solutions to Mean Curvature Flow with Uniform Bounds on the Mean Curvature and Its Gradient

    Priyamvada VishwamitraWider flows

    Topics Mean curvature flow · Shared terms “initial geometry”

  6. Quantitative uniqueness for mean curvature flow

    Tobias Holck Colding and William P. MinicozziWider flows

    Topics Singularities

  7. Singularity formation along the line bundle mean curvature flow

    Yu Hin Chan and Adam JacobWider flows

    Topics Long-time existence & convergence, Singularities

  8. Mean Curvature Flow of High Codimension in Complex Projective Space

    Artemis A. VogiatziWider flows

    Topics Long-time existence & convergence, 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. 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) Long-time existence & convergence A geometric flow either runs forever or stops because something blows up. Extension criteria say which quantity must blow up at a finite singular… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982)

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