Paper · Feb 19, 2019

Blow up dynamics for the hyperbolic vanishing mean curvature flow of surfaces asymptotic to Simons cone

Hajer Bahouri, Alaa Marachli, Galina Perelman

math.APWider flowsarXiv:1902.07027
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Abstract

In this article, we establish the existence of a family of hypersurfaces which evolve by the vanishing mean curvature flow in Minkowski space and which as tends to blow up towards a hypersurface which behaves like the Simons cone at infinity. This issue amounts to investigate the singularity formation for a second order quasilinear wave equation. Our constructive approach consists in proving the existence of finite time blow up solutions of this hyperbolic equation under the form , where is a stationary solution and an arbitrary large positive irrational number. Our approach roughly follows that of Krieger, Schlag and Tataru. However contrary to these works, the equation to be handled in this article is quasilinear. This induces a number of difficulties to face.

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Submitted
Feb 19, 2019
Subjects
math.AP
Mean curvature flowSingularities

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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.

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 Analysis of Velacuteazquez's solution to the mean curvature flow with a type II singularity (Siao-Hao Guo and Natasa Sesum, 2017)1Guo 2017 On the dynamics of formation of generic singularities of mean curvature flow (Zhou Gang, 2017)5Gang 2017 Mean curvature flow in homology and foliations of hyperbolic 3-manifolds (Marco A. M. Guaraco, Vanderson Lima and Franco Vargas Pallete, 2021)4Guaraco 2021 On the Structure of Singularities of Weak Mean Curvature Flows with Mean Curvature Bounds (Maxwell Stolarski, 2023)3Stolarski 2023 Blow Up of Compact Mean Curvature Flow Solutions with Bounded Mean Curvature (Zichang Liu, 2024)7Liu 2024 Singularity formations in Lagrangian mean curvature flow (Yang Li and Gábor Székelyhidi, 2024)8Li 2024 Mean curvature flow converging to an minimizing cone and its Hardt-Simon foliation (Jiuzhou Huang, 2025)6Huang 2025 Infinite-Time Singularities with Vanishing Mean Curvature for Lagrangian Mean Curvature Flow in Gibbons–Hawking Spaces (Ping-Hung Lee and Chung-Jun Tsai, 2026)2Lee 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Analysis of Velzquez's solution to the mean curvature flow with a type singularity

    Siao-Hao Guo and Natasa SesumWider flows

    Topics Singularities · Shared terms “Simons cone”

  2. Infinite-Time Singularities with Vanishing Mean Curvature for Lagrangian Mean Curvature Flow in Gibbons–Hawking Spaces

    Ping-Hung Lee and Chung-Jun TsaiWider flows

    Topics Singularities · Shared terms “vanishing mean”

  3. On the Structure of Singularities of Weak Mean Curvature Flows with Mean Curvature Bounds

    Maxwell StolarskiWider flows

    Topics Singularities · Shared terms “Simons cone”, “stationary”

  4. Mean curvature flow in homology and foliations of hyperbolic -manifolds

    Marco A. M. Guaraco, Vanderson Lima and Franco Vargas PalleteWider flows

    Topics Mean curvature flow · Shared terms “vanishing mean”

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

    Zhou GangWider flows

    Topics Singularities · Shared terms “formation”

  6. Mean curvature flow converging to an minimizing cone and its Hardt-Simon foliation

    Jiuzhou HuangWider flows

    Topics Singularities · Shared terms “Simon”

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

    Zichang LiuWider flows

    Topics Singularities · Shared terms “finite time”

  8. Singularity formations in Lagrangian mean curvature flow

    Yang Li and Gábor SzékelyhidiWider flows

    Topics Singularities · Shared terms “singularity formation”

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.