Paper · Dec 16, 2020

symmetry of the translating solitons of the mean curvature flow in

Jingze Zhu

math.DGWider flowsarXiv:2012.09319v2
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Abstract

In this paper, we prove that the translating solitons of the 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
Dec 16, 2020
Last revised
Jan 2, 2023 (v2)
Subjects
math.DG

Comments: Revised version, appeared in Annals of PDE

Mean curvature flowSelf-shrinkers & translatorsSingularities

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 Some results on translating solitons of the mean curvature flow (Jesús Pérez-García, 2016)3Pérez-García 2016 Complete translating solitons to the mean curvature flow in ℝ³ with nonnegative mean curvature (Joel Spruck and Ling Xiao, 2017)7Spruck 2017 Translating solitons of the mean curvature flow in the space ℍ²×ℝ (Antonio Bueno, 2018)8Bueno 2018 Notes on translating solitons for Mean Curvature Flow (David Hoffman, Tom Ilmanen, Francisco Martín et al., 2019)2Hoffman 2019 Convexity of 2-convex translating solitons to the mean curvature flow in ℝⁿ⁺¹ (Joel Spruck and Liming Sun, 2019)4Spruck 2019 A Zoo of Translating Solitons on a Parallel Light-like Direction in Minkowski 3-Space (Erdem Kocakuşaklı and Miguel Ortega, 2020)6Kocakuşaklı 2020 Rotational symmetry of uniformly 3-convex translating solitons of mean curvature flow in higher dimensions (Jingze Zhu, 2021)1Zhu 2021 Rigidity and Non-existence Results of λ-Translating Solitons (Li Xiang and Sun Jun, 2026)5Xiang 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Rotational symmetry of uniformly 3-convex translating solitons of mean curvature flow in higher dimensions

    Jingze ZhuWider flows

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

  2. Notes on translating solitons for Mean Curvature Flow

    David Hoffman, Tom Ilmanen, Francisco Martín et al.Wider flows

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

  3. Some results on translating solitons of the mean curvature flow

    Jesús Pérez-GarcíaWider flows

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

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

    Joel Spruck and Liming SunWider flows

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

  5. Rigidity and Non-existence Results of -Translating Solitons

    Li Xiang and Sun JunWider flows

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

  6. A Zoo of Translating Solitons on a Parallel Light-like Direction in Minkowski 3-Space

    Erdem Kocakuşaklı and Miguel OrtegaWider flows

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

  7. Complete translating solitons to the mean curvature flow in with nonnegative mean curvature

    Joel Spruck and Ling XiaoWider flows

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

  8. Translating solitons of the mean curvature flow in the space

    Antonio BuenoWider flows

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

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)

In the guidebook

  1. 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.

  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.