Paper · Mar 22, 2024

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

Miguel Ortega, Buse Yalçın

math.DGWider flowsarXiv:2403.15166
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Abstract

In this study, we deal with non-degenerate translators of the mean curvature flow in the well-known hyperbolic Einstein's static universe. We classify translators foliated by horospheres and rotationally invariant ones, both space-like and time-like. For space-like translators, we show a uniqueness theorem as well as a result to extend an isometry of the boundary of the domain to the whole translator, under simple conditions. As an application, we obtain a characterization of the the bowl when the boundary is a ball, and of certain translators foliated by horospheres whose boundary is a rectangle.

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Submitted
Mar 22, 2024
Subjects
math.DG
DOI
10.36890/iejg.1437356
Mean curvature flowSelf-shrinkers & translatorsStability & uniquenessRotational symmetry & warped products

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 Translators of the Gauss curvature flow (Muhittin Evren Aydin and Rafael López, 2022)7Aydin 2022 Translators to Higher Order Mean Curvature Flows in ℝⁿ×ℝ and ℍⁿ×ℝ (Ronaldo F. de Lima and Giuseppe Pipoli, 2022)4de Lima 2022 New Examples of Translating Solitons in Generalised Robertson-Walker Geometries (Diego Artacho, Marie-Amélie Lawn and Miguel Ortega, 2022)6Artacho 2022 Solitons to Mean Curvature Flow in the hyperbolic 3-space (R. F. de Lima, A. K. Ramos and J. P. dos Santos, 2023)2de Lima 2023 Rotationally symmetric translating solitons of fully nonlinear extrinsic geometric flows: Classification and Applications (José Torres Santaella, 2025)5Santaella 2025 Dynamical Stability of Translating Solitons to Mean Curvature Flow in Hyperbolic Space (Ronaldo F. de Lima and Álvaro K. Ramos, 2026)1de Lima 2026 Comparison Theorems for the Profile Curve Equation of Rotationally Symmetric Self-Shrinkers (Peng Peng, 2026)8Peng 2026 A Bernstein problem for translating solutions to the mean curvature flow (Juan Davila, Manuel del Pino and Juncheng Wei, 2026)3Davila 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Dynamical Stability of Translating Solitons to Mean Curvature Flow in Hyperbolic Space

    Ronaldo F. de Lima and Álvaro K. RamosWider flows

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

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

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

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

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

    Juan Davila, Manuel del Pino and Juncheng WeiWider flows

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

  4. Translators to Higher Order Mean Curvature Flows in and

    Ronaldo F. de Lima and Giuseppe PipoliWider flows

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

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

    José Torres SantaellaWider flows

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

  6. New Examples of Translating Solitons in Generalised Robertson-Walker Geometries

    Diego Artacho, Marie-Amélie Lawn and Miguel OrtegaWider flows

    Topics Self-shrinkers & translators · Same author Miguel Ortega

  7. Translators of the Gauss curvature flow

    Muhittin Evren Aydin and Rafael LópezWider flows

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

  8. Comparison Theorems for the Profile Curve Equation of Rotationally Symmetric Self-Shrinkers

    Peng PengWider flows

    Topics Rotational symmetry & warped products, Self-shrinkers & translators, Stability & uniqueness · 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.