Paper · Nov 26, 2025

Classification of Minimal Surfaces and Solitons to the Mean Curvature Flow in as Translation Surfaces

Tarcios Andrey Ferreira, João Paulo dos Santos

math.DGWider flowsarXiv:2511.21545v3
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Abstract

We consider the hyperbolic three-space in the half-space model endowed with a metric Lie group structure. In this setting, translation surfaces are defined as products of two curves and with respect to the Lie group operation. We investigate minimal surfaces and solitons to the mean curvature flow arising from specific types of products of these curves. In particular, we provide classification results for minimal surfaces, hyperbolic translators, and conformal solitons to the mean curvature flow.

Comments: We carefully revised the presentation of the manuscript and corrected typos throughout the text

Mean curvature flowSelf-shrinkers & translatorsSurfacesHomogeneous spacesMinimal & CMC surfaces

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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 Uniqueness of the [φ,e₃]-catenary cylinders by their asymptotic behaviour (A. L. Martínez-Triviño and J. P. dos Santos, 2021)7Martínez-Tri 2021 Conformal solitons for the mean curvature flow in hyperbolic space (Luciano Mari, Jose Danuso Rocha de Oliveira, Andreas Savas-Halilaj et al., 2023)6Mari 2023 Solitons to Mean Curvature Flow in the hyperbolic 3-space (R. F. de Lima, A. K. Ramos and J. P. dos Santos, 2023)4de Lima 2023 Rotators-Translators to Mean Curvature Flow in ℍ²×ℝ (Ronaldo F. de Lima, Álvaro K. Ramos and João Paulo dos Santos, 2024)5de Lima 2024 On the long-time limit of the mean curvature flow in closed manifolds (Alexander Mramor and Ao Sun, 2024)8Mramor 2024 The mean curvature flow of subgroups on Lie groups of dimension three (Romina M. Arroyo, Gabriela P. Ovando and Mariel Sáez, 2025)1Arroyo 2025 The Dirichlet problem for the minimal surface system on smooth domains (Caiyan Li and Hengyu Zhou, 2025)2Li 2025 Invariant λ-translators in the Heisenberg group (Antonio Bueno, 2026)3Bueno 2026
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  1. The mean curvature flow of subgroups on Lie groups of dimension three

    Romina M. Arroyo, Gabriela P. Ovando and Mariel SáezWider flows PDF

    Topics Minimal & CMC surfaces, Surfaces, Homogeneous spaces · Shared terms “Lie groups”

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

    Caiyan Li and Hengyu ZhouWider flows PDF

    Topics Minimal & CMC surfaces, Self-shrinkers & translators · Shared terms “half space”

  3. Invariant -translators in the Heisenberg group

    Antonio BuenoWider flows PDF

    Topics Minimal & CMC surfaces, Self-shrinkers & translators · Shared terms “translations”

  4. 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 · Same author João Paulo dos Santos

  5. 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 author João Paulo dos Santos

  6. Conformal solitons for the mean curvature flow in hyperbolic space

    Luciano Mari, Jose Danuso Rocha de Oliveira, Andreas Savas-Halilaj et al. PDF

    Topics Minimal & CMC surfaces, Self-shrinkers & translators · Shared terms “conformal solitons”, “space model”, “half space”

  7. Uniqueness of the -catenary cylinders by their asymptotic behaviour

    A. L. Martínez-Triviño and J. P. dos SantosWider flows PDF

    Topics Self-shrinkers & translators · Same author João Paulo dos Santos

  8. On the long-time limit of the mean curvature flow in closed manifolds

    Alexander Mramor and Ao SunWider flows PDF

    Topics Minimal & CMC surfaces

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) Homogeneous spaces On homogeneous spaces Ricci flow reduces to an ordinary differential equation on a finite-dimensional space of invariant metrics. Start with: Ricci flow of locally homogeneous geometries on closed manifolds (Isenberg et al., 1992)

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. 11A.13
    Homogeneous FlowsBook 11A · The Ricci Flow: Existence and Maximum Principles

    A computable laboratory: the flow on Thurston’s geometries.