Paper · May 31, 2016

Type-II singularities of two-convex immersed mean curvature flow

Theodora Bourni, Mat Langford

math.DGWider flowsarXiv:1605.09609
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As on arXiv

Abstract

We show that any strictly mean convex translator of dimension which admits a cylindrical estimate and a corresponding gradient estimate is rotationally symmetric. As a consequence, we deduce that any translating solution of the mean curvature flow which arises as a blow-up limit of a two-convex mean curvature flow of compact immersed hypersurfaces of dimension is rotationally symmetric. The proof is rather robust, and applies to a more general class of translator equations. As a particular application, we prove an analogous result for a class of flows of embedded hypersurfaces which includes the flow of two-convex hypersurfaces by the two-harmonic mean curvature.

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Submitted
May 31, 2016
Subjects
math.DG
Mean curvature flowSelf-shrinkers & translatorsSingularitiesRotational symmetry & warped products

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 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 On the existence of translating solutions of mean curvature flow in slab regions (Theodora Bourni, Mat Langford and Giuseppe Tinaglia, 2018)1Bourni 2018 Convex ancient solutions to curve shortening flow (Theodora Bourni, Mat Langford and Giuseppe Tinaglia, 2019)7Bourni 2019 Convex ancient solutions to mean curvature flow (Theodora Bourni, Mat Langford and Giuseppe Tinaglia, 2019)2Bourni 2019 The atomic structure of ancient grain boundaries (Theodora Bourni, Mat Langford and Giuseppe Tinaglia, 2020)4Bourni 2020 Rotational symmetry of uniformly 3-convex translating solitons of mean curvature flow in higher dimensions (Jingze Zhu, 2021)5Zhu 2021 Collapsing and noncollapsing in convex ancient mean curvature flow (Theodora Bourni, Mat Langford and Stephen Lynch, 2021)3Bourni 2021 Existence proofs for rotationally symmetric translating solutions to mean curvature flow (Hakar Raji and Oliver C. Schnürer, 2025)6Raji 2025 A Bernstein problem for translating solutions to the mean curvature flow (Juan Davila, Manuel del Pino and Juncheng Wei, 2026)8Davila 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. On the existence of translating solutions of mean curvature flow in slab regions

    Theodora Bourni, Mat Langford and Giuseppe TinagliaWider flows

    Topics Self-shrinkers & translators · Same authors Theodora Bourni, Mat Langford · Shared terms “translating solutions”, “convex translator”

  2. Convex ancient solutions to mean curvature flow

    Theodora Bourni, Mat Langford and Giuseppe TinagliaWider flows

    Topics Self-shrinkers & translators · Same authors Theodora Bourni, Mat Langford · Shared terms “convex translator”

  3. Collapsing and noncollapsing in convex ancient mean curvature flow

    Theodora Bourni, Mat Langford and Stephen LynchWider flows

    Topics Singularities · Same authors Theodora Bourni, Mat Langford · Shared terms “convex mean”

  4. The atomic structure of ancient grain boundaries

    Theodora Bourni, Mat Langford and Giuseppe TinagliaWider flows

    Topics Self-shrinkers & translators · Same authors Theodora Bourni, Mat Langford · Shared terms “translating solutions”, “analogous”

  5. Rotational symmetry of uniformly 3-convex translating solitons of mean curvature flow in higher dimensions

    Jingze ZhuWider flows

    Topics Rotational symmetry & warped products, Self-shrinkers & translators, Singularities · Shared terms “convex mean”, “mean convex”

  6. Existence proofs for rotationally symmetric translating solutions to mean curvature flow

    Hakar Raji and Oliver C. SchnürerWider flows

    Topics Rotational symmetry & warped products, Self-shrinkers & translators · Shared terms “translating solutions”, “rotationally symmetric”

  7. Convex ancient solutions to curve shortening flow

    Theodora Bourni, Mat Langford and Giuseppe TinagliaWider flows

    Topics Self-shrinkers & translators · Same authors Theodora Bourni, Mat Langford

  8. A Bernstein problem for translating solutions to the mean curvature flow

    Juan Davila, Manuel del Pino and Juncheng WeiWider flows

    Topics Rotational symmetry & warped products, Self-shrinkers & translators · Shared terms “translating solutions”, “mean convex”, “rotationally symmetric”

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) Self-shrinkers & translators Wider flows Self-similar solutions of mean curvature flow: shrinkers move by scaling, expanders by scaling outward, translators by sliding. Start with: Asymptotic behavior for singularities of the mean curvature flow (Huisken, 1990) 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) Rotational symmetry & warped products For rotationally symmetric metrics and warped products, Ricci flow reduces to a parabolic equation in one space variable. Start with: The ricci flow on radially symmetric R3 (Ivey, 1994)

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.