Paper · Mar 25, 2024

Blow Up of Compact Mean Curvature Flow Solutions with Bounded Mean Curvature

Zichang Liu

math.DGWider flowsarXiv:2403.16515
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Abstract

In 1994, Velázquez constructed a countable family of complete hypersurfaces flowing in by mean curvature, each of which develops a type II singularity at the origin in finite time. Later Guo and Sesum showed that for a non-empty subset of Velázquez's solutions, the mean curvature blows up near the origin, at a rate smaller than that of the second fundamental form; recently Stolarski proved another subset of these solutions has bounded mean curvature up to the singular time. In this paper, we follow their arguments to construct compact mean curvature flow solutions in with bounded mean curvature.

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Submitted
Mar 25, 2024
Subjects
math.DG, math.AP
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 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 Type II smoothing in mean curvature flow (Sigurd Angenent, Panagiota Daskalopoulos and Natasa Sesum, 2021)3Angenent 2021 Singularities of mean curvature flow with bounded mean curvature and Morse index (Yongheng Han, 2025)1Han 2025 The Bounded Diameter Conjecture and Sharp Geometric Estimates for Mean Curvature Flow (Yiqi Huang and Wenshuai Jiang, 2025)8Huang 2025 Mean curvature flow near a peanut solution (Sigurd Angenent, Panagiota Daskalopoulos and Natasa Sesum, 2025)5Angenent 2025 Morse resolution of mean curvature flows with cylindrical singularities (Richard H. Bamler, Felix Schulze and Lu Wang, 2026)4Bamler 2026 The mean curvature blows up at nondegenerate neck pinch singularities of Lagrangian mean curvature flow (Jason D. Lotay and Goncalo Oliveira, 2026)6Lotay 2026 Compact Mean Curvature Flows from Non-Compact Initial Data (Theodora Bourni, Nathan Burns and Marcus Flook, 2026)2Bourni 2026 Singularities of Mean Curvature Flow with Bounded Total Curvature (Yongheng Han and Zhen Wang, 2026)7Han 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 mean curvature and Morse index

    Yongheng HanWider flows

    Topics Singularities · Shared terms “bounded mean”, “singular time”

  2. Compact Mean Curvature Flows from Non-Compact Initial Data

    Theodora Bourni, Nathan Burns and Marcus FlookWider flows

    Topics Singularities · Shared terms “compact mean”, “construct compact”, “finite time”

  3. Type II smoothing in mean curvature flow

    Sigurd Angenent, Panagiota Daskalopoulos and Natasa SesumWider flows

    Topics Singularities · Shared terms “Velazquez”, “Stolarski”, “fundamental form”

  4. Morse resolution of mean curvature flows with cylindrical singularities

    Richard H. Bamler, Felix Schulze and Lu WangWider flows

    Topics Singularities · Shared terms “compact mean”

  5. Mean curvature flow near a peanut solution

    Sigurd Angenent, Panagiota Daskalopoulos and Natasa SesumWider flows

    Topics Singularities · Shared terms “Velazquez”, “finite time”

  6. The mean curvature blows up at nondegenerate neck pinch singularities of Lagrangian mean curvature flow

    Jason D. Lotay and Goncalo OliveiraWider flows

    Topics Singularities · Shared terms “curvature blow”

  7. Singularities of Mean Curvature Flow with Bounded Total Curvature

    Yongheng Han and Zhen WangWider flows

    Topics Singularities · Shared terms “singular time”

  8. The Bounded Diameter Conjecture and Sharp Geometric Estimates for Mean Curvature Flow

    Yiqi Huang and Wenshuai JiangWider flows

    Topics Singularities · Shared terms “fundamental form”, “singular time”

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)

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.