Paper · Dec 12, 2016

Convexity and the Dirichlet problem of translating mean curvature flows

Li Ma

math.DGWider flowsarXiv:1612.03576
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Abstract

In this work, we propose a new evolving geometric flow (called translating mean curvature flow) for the translating solitons of hypersurfaces in . We study the basic properties, such as positivity preserving property, of the translating mean curvature flow. The Dirichlet problem for the graphical translating mean curvature flow is studied and the global existence of the flow and the convergence property are also considered.

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Submitted
Dec 12, 2016
Subjects
math.DG, math.AP

Comments: 10 pages

Mean curvature flowSelf-shrinkers & translatorsLong-time existence & convergence

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)7Pérez-García 2016 Complete translating solitons to the mean curvature flow in ℝ³ with nonnegative mean curvature (Joel Spruck and Ling Xiao, 2017)3Spruck 2017 Notes on translating solitons for Mean Curvature Flow (David Hoffman, Tom Ilmanen, Francisco Martín et al., 2019)6Hoffman 2019 Convexity of 2-convex translating solitons to the mean curvature flow in ℝⁿ⁺¹ (Joel Spruck and Liming Sun, 2019)1Spruck 2019 SO(2) symmetry of the translating solitons of the mean curvature flow in ℝ⁴ (Jingze Zhu, 2020)2Zhu 2020 Graphical translating solitons for the inverse mean curvature flow and isoparametric functions (Tomoki Fujii, 2022)8Fujii 2022 Rigidity results for mean curvature flow graphical translators moving in non-graphical direction (John Man Shun Ma, Yuan Shyong Ooi and Juncheol Pyo, 2022)5Ma 2022 The Dirichlet problem for the minimal surface system on smooth domains (Caiyan Li and Hengyu Zhou, 2025)4Li 2025
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 · Shared terms “graphical translating”, “translating solitons”

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

    Jingze ZhuWider flows

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

  3. 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 “graphical translating”, “translating solitons”

  4. The Dirichlet problem for the minimal surface system on smooth domains

    Caiyan Li and Hengyu ZhouWider flows

    Topics Self-shrinkers & translators, Long-time existence & convergence · Shared terms “Dirichlet”

  5. Rigidity results for mean curvature flow graphical translators moving in non-graphical direction

    John Man Shun Ma, Yuan Shyong Ooi and Juncheol PyoWider flows

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

  6. 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”

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

    Jesús Pérez-GarcíaWider flows

    Topics Self-shrinkers & translators · Shared terms “translating solitons”, “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) Long-time existence & convergence A geometric flow either runs forever or stops because something blows up. Extension criteria say which quantity must blow up at a finite singular… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982)

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.