Paper · Mar 3, 2017

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

Joel Spruck, Ling Xiao

math.DGWider flowsarXiv:1703.01003v3
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Abstract

We prove that any complete immersed two-sided mean convex translating soliton for the mean curvature flow is convex. As a corollary it follows that an entire mean convex graphical translating soliton in is the axisymmetric "bowl soliton". We also show that if the mean curvature of tends to zero at infinity, then can be represented as an entire graph and so is the "bowl soliton". Finally we classify all locally strictly convex graphical translating solitons defined over strip regions.

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Submitted
Mar 3, 2017
Last revised
May 29, 2018 (v3)
Subjects
math.DG
Mean curvature flowSelf-shrinkers & translators

Its neighbourhood

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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 Graphical Translators for Mean Curvature Flow (David Hoffman, Tom Ilmanen, Francisco Martin et al., 2018)7Hoffman 2018 Notes on translating solitons for Mean Curvature Flow (David Hoffman, Tom Ilmanen, Francisco Martín et al., 2019)4Hoffman 2019 Convexity of 2-convex translating solitons to the mean curvature flow in ℝⁿ⁺¹ (Joel Spruck and Liming Sun, 2019)1Spruck 2019 The prescribed curvature problem for entire hypersurfaces in Minkowski space (Changyu Ren, Zhizhang Wang and Ling Xiao, 2020)2Ren 2020 SO(2) symmetry of the translating solitons of the mean curvature flow in ℝ⁴ (Jingze Zhu, 2020)3Zhu 2020 Rotational symmetry of uniformly 3-convex translating solitons of mean curvature flow in higher dimensions (Jingze Zhu, 2021)6Zhu 2021 Graphical translating solitons for the inverse mean curvature flow and isoparametric functions (Tomoki Fujii, 2022)8Fujii 2022 A Bernstein problem for translating solutions to the mean curvature flow (Juan Davila, Manuel del Pino and Juncheng Wei, 2026)5Davila 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Convexity of 2-convex translating solitons to the mean curvature flow in

    Joel Spruck and Liming SunWider flows

    Topics Self-shrinkers & translators · Same author Joel Spruck · Shared terms “translating solitons”, “convex translating”, “graphical translating”

  2. The prescribed curvature problem for entire hypersurfaces in Minkowski space

    Changyu Ren, Zhizhang Wang and Ling XiaoWider flows

    Topics Self-shrinkers & translators · Same author Ling Xiao · Shared terms “translating solitons”, “strictly convex”

  3. symmetry of the translating solitons of the mean curvature flow in

    Jingze ZhuWider flows

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

  4. 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”, “complete translating”

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

    Juan Davila, Manuel del Pino and Juncheng WeiWider flows

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

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

    Jingze ZhuWider flows

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

  7. Graphical Translators for Mean Curvature Flow

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

    Topics Self-shrinkers & translators · Shared terms “complete translating”, “graphical”

  8. Graphical translating solitons for the inverse mean curvature flow and isoparametric functions

    Tomoki FujiiWider flows

    Topics Self-shrinkers & translators · Shared terms “graphical translating”, “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)

In the guidebook

  1. 6A.10
    Curve Shortening and the First Geometric FlowsBook 6A · The Heat Equation and Its Relatives

    Shrinking circles, grim reapers, neckpinches and Huisken’s monotonicity, seen in the plane.