Paper · Apr 21, 2026

Translating solitons to higher order mean curvature flows in Riemannian products

Jorge Herbert Soares de Lira, Rafael Rocha de Farias

math.DGWider flowsarXiv:2604.18974
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Abstract

In this paper we prove existence and classification results for translating solitons defined as initial conditions for higher order mean curvature flows that are invariant by translations in warped product manifolds . Here, is a Cartan-Hadamard manifold endowed with a rotationally symmetric metric and is a radial function defined in . In this setting, the higher order mean curvature flow is, up to a change of time parameter, given by translations along the factor in the warped product. This setting encompasses the cases of translating solitons in , and studied in recent papers. In particular we prove the existence of families of bowl-type and catenoid-type translating solitons under mild assumptions about the curvature of the warped product. We also describe the asymptotic behavior for those solitons in terms of the geometry at infinity of . Our assumptions about the ambient metric allow us to control the higher order mean curvature of cylinders and to use them as barriers.

Comments: 44 pages

Mean curvature flowSelf-shrinkers & translatorsRotational 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 Rotationally symmetric translating solutions to extrinsic geometric flows (Sathyanarayanan Rengaswami, 2021)7Rengaswami 2021 Translators to Higher Order Mean Curvature Flows in ℝⁿ×ℝ and ℍⁿ×ℝ (Ronaldo F. de Lima and Giuseppe Pipoli, 2022)1de Lima 2022 Mean curvature flows of graphs sliding off to infinity in warped product manifolds (Naotoshi Fujihara, 2023)6Fujihara 2023 Isotropy invariant graphical mean curvature flows in warped products (Naotoshi Fujihara and Naoyuki Koike, 2024)8Fujihara 2024 Existence proofs for rotationally symmetric translating solutions to mean curvature flow (Hakar Raji and Oliver C. Schnürer, 2025)3Raji 2025 Rotationally symmetric translating solitons of fully nonlinear extrinsic geometric flows: Classification and Applications (José Torres Santaella, 2025)4Santaella 2025 Dynamical Stability of Translating Solitons to Mean Curvature Flow in Hyperbolic Space (Ronaldo F. de Lima and Álvaro K. Ramos, 2026)2de 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. Translators to Higher Order Mean Curvature Flows in and

    Ronaldo F. de Lima and Giuseppe PipoliWider flows PDF

    Topics Self-shrinkers & translators · Shared terms “order mean”, “higher order”, “catenoid type”

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

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

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

  3. 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 symmetric”

  4. 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 · Shared terms “translating solitons”, “bowl type”, “rotationally symmetric”

  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 · Shared terms “translating solitons”, “rotationally symmetric”

  6. Mean curvature flows of graphs sliding off to infinity in warped product manifolds

    Naotoshi FujiharaWider flows PDF

    Topics Rotational symmetry & warped products · Shared terms “higher order”

  7. Rotationally symmetric translating solutions to extrinsic geometric flows

    Sathyanarayanan RengaswamiWider flows PDF

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

  8. Isotropy invariant graphical mean curvature flows in warped products

    Naotoshi Fujihara and Naoyuki KoikeWider flows PDF

    Topics Rotational symmetry & warped products

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) 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.