Paper · Jan 29, 2026

Dynamical stability of various convex graphical translators

Junyoung Park

math.DGWider flowsarXiv:2601.22368v2
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Abstract

In the first part of the paper, we prove the existence of longtime solution to mean curvature flow starting from a graph of a continuous function defined over a slab. Then, we establish dynamical stability results for various types of graphical translators to mean curvature flow, namely the grim reaper, two dimensional graphical translators, and asymptotically cylindrical translators.

Comments: Comments welcome. Added a corollary imposing an entropy bound assumption instead of mean convexity in asymptotically cylindrical case

Mean curvature flowSelf-shrinkers & translatorsSurfacesStability & uniquenessLong-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 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)2Cesaroni 2021 Rigidity results for mean curvature flow graphical translators moving in non-graphical direction (John Man Shun Ma, Yuan Shyong Ooi and Juncheol Pyo, 2022)1Ma 2022 Translators to Higher Order Mean Curvature Flows in ℝⁿ×ℝ and ℍⁿ×ℝ (Ronaldo F. de Lima and Giuseppe Pipoli, 2022)6de Lima 2022 Solitons to Mean Curvature Flow in the hyperbolic 3-space (R. F. de Lima, A. K. Ramos and J. P. dos Santos, 2023)7de Lima 2023 The class of grim reapers in ℍ²×ℝ (Antonio Bueno and Rafael López, 2024)4Bueno 2024 Rotationally symmetric translating solitons of fully nonlinear extrinsic geometric flows: Classification and Applications (José Torres Santaella, 2025)8Santaella 2025 Dynamical Stability of Translating Solitons to Mean Curvature Flow in Hyperbolic Space (Ronaldo F. de Lima and Álvaro K. Ramos, 2026)3de Lima 2026 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. Rigidity results for mean curvature flow graphical translators moving in non-graphical direction

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

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

  2. Graphical translators for anisotropic and crystalline mean curvature flow

    Annalisa Cesaroni, Heiko Kroener and Matteo NovagaWider flows PDF

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

  3. Dynamical Stability of Translating Solitons to Mean Curvature Flow in Hyperbolic Space

    Ronaldo F. de Lima and Álvaro K. RamosWider flows PDF

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

  4. The class of grim reapers in

    Antonio Bueno and Rafael LópezWider flows PDF

    Topics Self-shrinkers & translators · Shared terms “grim reaper”

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

    Juan Davila, Manuel del Pino and Juncheng WeiWider flows PDF

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

  6. Translators to Higher Order Mean Curvature Flows in and

    Ronaldo F. de Lima and Giuseppe PipoliWider flows PDF

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

  7. Solitons to Mean Curvature Flow in the hyperbolic 3-space

    R. F. de Lima, A. K. Ramos and J. P. dos SantosWider flows PDF

    Topics Self-shrinkers & translators, Stability & uniqueness

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

    José Torres SantaellaWider flows PDF

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

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) Surfaces On surfaces Ricci flow is conformal and scalar. It gives a heat-flow proof of the uniformization theorem: every closed surface converges to constant… Start with: The Ricci flow on surfaces (Hamilton, 1988) 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. 5A.5
    Uniformization and the Two-Dimensional Ricci FlowBook 5A · Complex Analysis and Conformal Geometry

    Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.

  2. 11A.12
    Ricci Flow on SurfacesBook 11A · The Ricci Flow: Existence and Maximum Principles

    Uniformization by Ricci flow, and Hamilton’s entropy.

  3. 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.