Paper · Mar 30, 2021

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

Jingze Zhu

math.DGWider flowsarXiv:2103.16382
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As on arXiv

Abstract

In this paper, we generalize a previous result to higher dimension. We prove that uniformly 3-convex translating solitons of mean curvature flow in which arise as blow up limit of embedded, mean convex mean curvature flow must have symmetry.

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Submitted
Mar 30, 2021
Subjects
math.DG

Comments: arXiv admin note: substantial text overlap with arXiv:2012.09319

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 Type-II singularities of two-convex immersed mean curvature flow (Theodora Bourni and Mat Langford, 2016)7Bourni 2016 Rotational Symmetry of Asymptotically Conical Mean Curvature Flow Self-Expanders (Frederick Tsz-Ho Fong and Peter McGrath, 2016)5Fong 2016 Uniqueness of convex ancient solutions to mean curvature flow in higher dimensions (S. Brendle and K. Choi, 2018)4Brendle 2018 Convexity of 2-convex translating solitons to the mean curvature flow in ℝⁿ⁺¹ (Joel Spruck and Liming Sun, 2019)2Spruck 2019 Simply connected translating solitons contained in slabs (Francesco Chini, 2019)8Chini 2019 SO(2) symmetry of the translating solitons of the mean curvature flow in ℝ⁴ (Jingze Zhu, 2020)1Zhu 2020 Rotational symmetry of ancient solutions to fully nonlinear curvature flows (A. Cogo, S. Lynch and O. Vičánek Martínez, 2023)6Cogo 2023 A Bernstein problem for translating solutions to the mean curvature flow (Juan Davila, Manuel del Pino and Juncheng Wei, 2026)3Davila 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. symmetry of the translating solitons of the mean curvature flow in

    Jingze ZhuWider flows

    Topics Self-shrinkers & translators, Singularities · Same author Jingze Zhu · Shared terms “translating solitons”, “convex mean”, “mean convex”

  2. Convexity of 2-convex translating solitons to the mean curvature flow in

    Joel Spruck and Liming SunWider flows

    Topics Self-shrinkers & translators · Shared terms “convex translating”, “translating solitons”

  3. 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 “convex translating”, “higher dimensions”, “translating solitons”

  4. Uniqueness of convex ancient solutions to mean curvature flow in higher dimensions

    S. Brendle and K. ChoiWider flows

    Topics Rotational symmetry & warped products, Self-shrinkers & translators · Shared terms “higher dimensions”, “translating solitons”

  5. Rotational Symmetry of Asymptotically Conical Mean Curvature Flow Self-Expanders

    Frederick Tsz-Ho Fong and Peter McGrathWider flows

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

  6. Rotational symmetry of ancient solutions to fully nonlinear curvature flows

    A. Cogo, S. Lynch and O. Vičánek MartínezWider flows

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

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

    Theodora Bourni and Mat LangfordWider flows

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

  8. Simply connected translating solitons contained in slabs

    Francesco ChiniWider flows

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

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)

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  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.