Paper · Oct 7, 2022

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

John Man Shun Ma, Yuan Shyong Ooi, Juncheol Pyo

math.DGWider flowsarXiv:2210.03707
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Abstract

In this paper, we study the rigidity results of complete graphical translating hypersurfaces when the translating direction is not in the graphical direction. We proved that any entire graphical translating surface in the translating direction not parallel to the graphical one is flat if either the translating surface is mean convex or the entropy of the translating surface is smaller than . For higher dimensional case, we show that the same conclusion holds if the graphical translating hypersurface satisfies certain growth condition.

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Submitted
Oct 7, 2022
Subjects
math.DG
Mean curvature flowSelf-shrinkers & translatorsStability & uniqueness

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 2021, 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 anisotropic and crystalline mean curvature flow (Annalisa Cesaroni, Heiko Kroener and Matteo Novaga, 2021)3Cesaroni 2021 Graphical translating solitons for the mean curvature flow and isoparametric functions (Tomoki Fujii, 2021)4Fujii 2021 Graphical translating solitons for the inverse mean curvature flow and isoparametric functions (Tomoki Fujii, 2022)8Fujii 2022 K^α-translators of offset surfaces (Burcu Bektaş Demirci, Ferdağ Kahraman Aksoyak and Murat Babaarslan, 2024)7Demirci 2024 Rotationally symmetric translating solitons of fully nonlinear extrinsic geometric flows: Classification and Applications (José Torres Santaella, 2025)6Santaella 2025 Half-space theorems for translating solitons of the r-mean curvature flow (Hilário Alencar, G. Pacelli Bessa and Gregório Silva Neto, 2026)5Alencar 2026 Dynamical stability of various convex graphical translators (Junyoung Park, 2026)2Park 2026 A Bernstein problem for translating solutions to the mean curvature flow (Juan Davila, Manuel del Pino and Juncheng Wei, 2026)1Davila 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. A Bernstein problem for translating solutions to the mean curvature flow

    Juan Davila, Manuel del Pino and Juncheng WeiWider flows

    Topics Self-shrinkers & translators, Stability & uniqueness · Shared terms “graphical translators”, “entire graphical”, “graphical translating”

  2. Dynamical stability of various convex graphical translators

    Junyoung ParkWider flows

    Topics Self-shrinkers & translators, Stability & uniqueness · Shared terms “graphical translators”

  3. Graphical translators for anisotropic and crystalline mean curvature flow

    Annalisa Cesaroni, Heiko Kroener and Matteo NovagaWider flows

    Topics Self-shrinkers & translators, Stability & uniqueness · Shared terms “graphical translators”

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

    Tomoki FujiiWider flows

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

  5. Half-space theorems for translating solitons of the r-mean curvature flow

    Hilário Alencar, G. Pacelli Bessa and Gregório Silva NetoWider flows

    Topics Self-shrinkers & translators · Shared terms “growth condition”, “direction”

  6. Rotationally symmetric translating solitons of fully nonlinear extrinsic geometric flows: Classification and Applications

    José Torres SantaellaWider flows

    Topics Self-shrinkers & translators, Stability & uniqueness · Shared terms “graphical translators”, “entire graphical”

  7. -translators of offset surfaces

    Burcu Bektaş Demirci, Ferdağ Kahraman Aksoyak and Murat BabaarslanWider flows

    Topics Self-shrinkers & translators · Shared terms “translators moving”, “direction”, “parallel”

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

    Tomoki FujiiWider flows

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

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. 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) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005)

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.