Paper · May 14, 2021

A numerical stability analysis of mean curvature flow of noncompact hypersurfaces with Type-II curvature blowup

David Garfinkle, James Isenberg, Dan Knopf, Haotian Wu

math.DGWider flowsarXiv:2105.06613
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Abstract

We present a numerical study of the local stability of mean curvature flow of rotationally symmetric, complete noncompact hypersurfaces with Type-II curvature blowup. Our numerical analysis employs a novel overlap method that constructs "numerically global" (i.e., with spatial domain arbitrarily large but finite) flow solutions with initial data covering analytically distinct regions. Our numerical results show that for certain prescribed families of perturbations, there are two classes of initial data that lead to distinct behaviors under mean curvature flow. Firstly, there is a "near" class of initial data which lead to the same singular behaviour as an unperturbed solution; in particular, the curvature at the tip of the hypersurface blows up at a Type-II rate no slower than . Secondly, there is a "far" class of initial data which lead to solutions developing a local Type-I nondegenerate neckpinch under mean curvature flow. These numerical findings further suggest the existence of a "critical" class of initial data which conjecturally lead to mean curvature flow of noncompact hypersurfaces forming local Type-II degenerate neckpinches with the highest curvature blowup rate strictly slower than .

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Submitted
May 14, 2021
Subjects
math.DG, math.AP, math.NA
DOI
10.1088/1361-6544/ac15a9

Comments: 23 pages, 11 figures. Comments are welcome!

Mean curvature flowSingularitiesStability & uniquenessBoundaries & non-compactRotational symmetry & warped productsNumerics & visualization

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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 Mean curvature flow of noncompact hypersurfaces with Type-II curvature blow-up (James Isenberg and Haotian Wu, 2016)3Isenberg 2016 Non-Kahler Ricci flow singularities modeled on Kahler-Ricci solitons (James Isenberg, Dan Knopf and Natasa Sesum, 2017)8Isenberg 2017 Mean curvature flow of noncompact hypersurfaces with Type-II curvature blow-up. II (James Isenberg, Haotian Wu and Zhou Zhang, 2019)2Isenberg 2019 On the precise asymptotics of Type-IIb solutions to mean curvature flow (James Isenberg, Haotian Wu and Zhou Zhang, 2020)4Isenberg 2020 Singularity formation of complete Ricci flow solutions (Timothy Carson, James Isenberg, Dan Knopf et al., 2020)6Carson 2020 A numerical stability analysis of mean curvature flow of noncompact hypersurfaces with Type-II curvature blowup: II (David Garfinkle, James Isenberg, Dan Knopf et al., 2022)1Garfinkle 2022 Asymptotic behavior of unstable perturbations of the Fubini-Study metric in Ricci flow (David Garfinkle, James Isenberg, Dan Knopf et al., 2024)5Garfinkle 2024 Local singularities of compact multiply warped Ricci flow solutions (James Isenberg, Dan Knopf, Zilu Ma et al., 2025)7Isenberg 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. A numerical stability analysis of mean curvature flow of noncompact hypersurfaces with Type-II curvature blowup: II

    David Garfinkle, James Isenberg, Dan Knopf et al.Wider flows

    Topics Numerics & visualization, Rotational symmetry & warped products, Boundaries & non-compact · Same authors David Garfinkle, James Isenberg, Dan Knopf · Shared terms “curvature blowup”, “numerical stability”, “noncompact hypersurfaces”

  2. Mean curvature flow of noncompact hypersurfaces with Type-II curvature blow-up. II

    James Isenberg, Haotian Wu and Zhou ZhangWider flows

    Topics Rotational symmetry & warped products, Boundaries & non-compact, Singularities · Same authors James Isenberg, Haotian Wu · Shared terms “noncompact hypersurfaces”, “highest curvature”, “complete noncompact”

  3. Mean curvature flow of noncompact hypersurfaces with Type-II curvature blow-up

    James Isenberg and Haotian WuWider flows

    Topics Rotational symmetry & warped products, Boundaries & non-compact, Singularities · Same authors James Isenberg, Haotian Wu · Shared terms “noncompact hypersurfaces”, “highest curvature”, “rotationally symmetric”

  4. On the precise asymptotics of Type-IIb solutions to mean curvature flow

    James Isenberg, Haotian Wu and Zhou ZhangWider flows

    Topics Rotational symmetry & warped products, Boundaries & non-compact, Singularities · Same authors James Isenberg, Haotian Wu · Shared terms “highest curvature”, “rotationally symmetric”

  5. Asymptotic behavior of unstable perturbations of the Fubini-Study metric in Ricci flow

    David Garfinkle, James Isenberg, Dan Knopf et al.

    Topics Numerics & visualization, Singularities, Stability & uniqueness · Same authors David Garfinkle, James Isenberg, Dan Knopf · Shared terms “numerical”, “perturbation”

  6. Singularity formation of complete Ricci flow solutions

    Timothy Carson, James Isenberg, Dan Knopf et al.

    Topics Boundaries & non-compact, Singularities, Stability & uniqueness · Same authors James Isenberg, Dan Knopf

  7. Local singularities of compact multiply warped Ricci flow solutions

    James Isenberg, Dan Knopf, Zilu Ma et al.

    Topics Rotational symmetry & warped products, Boundaries & non-compact, Singularities · Same authors James Isenberg, Dan Knopf · Shared terms “complete noncompact”

  8. Non-Kahler Ricci flow singularities modeled on Kahler-Ricci solitons

    James Isenberg, Dan Knopf and Natasa Sesum

    Topics Singularities, Stability & uniqueness · Same authors James Isenberg, Dan Knopf

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) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005) Boundaries & non-compact Ricci flow beyond closed manifolds: complete non-compact spaces, manifolds with boundary, and asymptotically flat or hyperbolic ends. Start with: Deforming the metric on complete Riemannian manifolds (Shi, 1989)

In the guidebook

  1. 6D.11
    Computing Geometric FlowsBook 6D · Diffusions, Finance and Computation

    Numerical curve shortening and mean curvature flow, level sets and phase fields, discrete Ricci flow on surfaces and in conformal mapping, rotationally symmetric neckpinches, and what computation has taught about singularities.

  2. 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.

  3. 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.