Paper · Feb 2, 2026

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

Ronaldo F. de Lima, Álvaro K. Ramos

math.DGWider flowsarXiv:2602.02424
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Abstract

We develop the theory of translating solitons for the Mean Curvature Flow (MCF) in hyperbolic space of dimension . More specifically, we establish that horospheres are dynamically stable as radial graphical solutions to MCF. To that end, we construct rotationally invariant translators analogous to the winglike solitons introduced by Clutterbuck, Schnürer and Schulze, which serve as barriers in an argument based on White's avoidance principle and the strong maximum principle for parabolic PDEs.

Comments: 13 pages, 3 figures. Comments are welcome

Mean curvature flowSelf-shrinkers & translatorsStability & uniquenessRotational symmetry & warped products

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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 Translators to Higher Order Mean Curvature Flows in ℝⁿ×ℝ and ℍⁿ×ℝ (Ronaldo F. de Lima and Giuseppe Pipoli, 2022)2de 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)1de Lima 2023 Rotators-Translators to Mean Curvature Flow in ℍ²×ℝ (Ronaldo F. de Lima, Álvaro K. Ramos and João Paulo dos Santos, 2024)3de Lima 2024 Translators of the Mean Curvature Flow in Hyperbolic Einstein's Static Universe (Miguel Ortega and Buse Yalçın, 2024)4Ortega 2024 Existence proofs for rotationally symmetric translating solutions to mean curvature flow (Hakar Raji and Oliver C. Schnürer, 2025)8Raji 2025 Rotationally symmetric translating solitons of fully nonlinear extrinsic geometric flows: Classification and Applications (José Torres Santaella, 2025)7Santaella 2025 Translating solitons to higher order mean curvature flows in Riemannian products (Jorge Herbert Soares de Lira and Rafael Rocha de Farias, 2026)6de Lira 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. 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 · Same authors Ronaldo F. de Lima, Álvaro K. Ramos · Shared terms “MCF”, “translating solitons”, “horosphere”

  2. Translators to Higher Order Mean Curvature Flows in and

    Ronaldo F. de Lima and Giuseppe PipoliWider flows PDF

    Topics Self-shrinkers & translators, Stability & uniqueness · Same author Ronaldo F. de Lima · Shared terms “MCF”, “Clutterbuck”, “Schnürer”

  3. Rotators-Translators to Mean Curvature Flow in

    Ronaldo F. de Lima, Álvaro K. Ramos and João Paulo dos SantosWider flows PDF

    Topics Self-shrinkers & translators · Same authors Ronaldo F. de Lima, Álvaro K. Ramos

  4. Translators of the Mean Curvature Flow in Hyperbolic Einstein's Static Universe

    Miguel Ortega and Buse YalçınWider flows PDF

    Topics Rotational symmetry & warped products, Self-shrinkers & translators, Stability & uniqueness · Shared terms “horosphere”, “rotationally invariant”

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

    Juan Davila, Manuel del Pino and Juncheng WeiWider flows PDF

    Topics Rotational symmetry & warped products, Self-shrinkers & translators, Stability & uniqueness · Shared terms “translating solitons”, “graphical”, “rotationally”

  6. Translating solitons to higher order mean curvature flows in Riemannian products

    Jorge Herbert Soares de Lira and Rafael Rocha de FariasWider flows PDF

    Topics Rotational symmetry & warped products, Self-shrinkers & translators · Shared terms “translating solitons”, “rotationally”

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

    José Torres SantaellaWider flows PDF

    Topics Rotational symmetry & warped products, Self-shrinkers & translators, Stability & uniqueness · Shared terms “translating solitons”, “rotationally”, “graphical”

  8. Existence proofs for rotationally symmetric translating solutions to mean curvature flow

    Hakar Raji and Oliver C. SchnürerWider flows PDF

    Topics Rotational symmetry & warped products, Self-shrinkers & translators · Shared terms “rotationally”

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) Rotational symmetry & warped products For rotationally symmetric metrics and warped products, Ricci flow reduces to a parabolic equation in one space variable. Start with: The ricci flow on radially symmetric R3 (Ivey, 1994)

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.