Paper · Jan 7, 2017

Analysis of Velzquez's solution to the mean curvature flow with a type singularity

Siao-Hao Guo, Natasa Sesum

math.DGWider flowsarXiv:1701.01835v2
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Abstract

J.J.L. Velzquez in 1994 used the degree theory to show that there is a perturbation of Simons' cone, starting from which the mean curvature flow develops a type singularity at the origin. He also showed that under a proper time-dependent rescaling of the solution around the origin, the rescaled flow converges in the sense to a minimal hypersurface which is tangent to Simons' cone at infinity. In this paper, we prove that the rescaled flow actually converges locally smoothly to the minimal hypersurface, which appears to be the singularity model of the type singularity. In addition, we show that the mean curvature of the solution blows up near the origin at a rate which is smaller than that of the second fundamental form.

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Submitted
Jan 7, 2017
Last revised
Jul 12, 2017 (v2)
Subjects
math.DG

Comments: Added a footnote

Mean curvature flowSingularitiesMinimal & CMC surfaces

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    Topics Minimal & CMC surfaces, Singularities · Shared terms “minimal hypersurfaces”

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    Topics Minimal & CMC surfaces, Singularities · Shared terms “minimal hypersurfaces”

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    Topics Minimal & CMC surfaces · Shared terms “minimal hypersurfaces”

  6. Mean curvature flow with multiplicity convergence in closed manifolds

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    Topics Minimal & CMC surfaces · Shared terms “minimal hypersurfaces”

  7. Type II smoothing in mean curvature flow

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    Topics Singularities · Same author Natasa Sesum · Shared terms “fundamental form”

  8. Mean curvature flow near a peanut solution

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    Topics Singularities · Same author Natasa Sesum · Shared terms “perturbation”, “rescaled”

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) Minimal & CMC surfaces Minimal surfaces are critical points of area, with zero mean curvature; constant mean curvature (CMC) surfaces are critical for area at fixed volume. Start with: On the structure of manifolds with positive scalar curvature (Schoen et al., 1979)

In the guidebook

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    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.