Paper · Mar 13, 2020

Existence of Mean Curvature Flow Singularities with Bounded Mean Curvature

Maxwell Stolarski

math.DGWider flowsarXiv:2003.06383
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Abstract

In [Vel94], Velazquez constructed a countable collection of mean curvature flow solutions in in every dimension . Each of these solutions becomes singular in finite time at which time the second fundamental form blows up. In contrast, we confirm here that, in every dimension , a nontrivial subset of these solutions has uniformly bounded mean curvature.

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Submitted
Mar 13, 2020
Subjects
math.DG, math.AP

Comments: 44 pages, comments welcome

Mean curvature flowSingularities

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  1. Existence of Lagrangian Mean Curvature Flow Singularities with Bounded Mean Curvature

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    Topics Singularities · Same author Maxwell Stolarski · Shared terms “bounded mean”, “uniformly bounded”, “finite time”

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    Topics Singularities · Shared terms “bounded mean”, “Velazquez constructed”, “countable”

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    Topics Singularities · Shared terms “bounded mean”

  4. Type II smoothing in mean curvature flow

    Sigurd Angenent, Panagiota Daskalopoulos and Natasa SesumWider flows

    Topics Singularities · Shared terms “Velazquez”, “uniformly bounded”, “fundamental form”

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

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    Topics Singularities · Same author Maxwell Stolarski

  6. Finite-Time Singularities of Lagrangian Mean Curvature Flow with Quantitatively Precise Dynamics

    Maxwell Stolarski and Wei-Bo SuWider flows

    Topics Singularities · Same author Maxwell Stolarski · Shared terms “finite time”

  7. Spectral Analysis for Finite-Time Singularities of Lagrangian Mean Curvature Flow

    Maxwell Stolarski and Wei-Bo SuWider flows

    Topics Singularities · Same author Maxwell Stolarski · Shared terms “finite time”

  8. Closed Ricci Flows with Singularities Modeled on Asymptotically Conical Shrinkers

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    Topics Singularities · Same author Maxwell Stolarski · Shared terms “finite 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 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

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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.