Paper · Nov 26, 2022

New Examples of Translating Solitons in Generalised Robertson-Walker Geometries

Diego Artacho, Marie-Amélie Lawn, Miguel Ortega

math.DGWider flowsarXiv:2211.14529v2

As on arXiv

Abstract

Translators can be regarded as submanifolds which satisfy the mean curvature flow equation when evolving by translations along a distinguished vector field of the ambient space. We study translators in Generalised Robertson-Walker spacetimes, due to their importance as Lorentzian manifolds, and because they admit a natural conformal Killing timelike vector field carrying substantial geometric information, which will play the role of this translating vector field. We identify three one-parameter families of warping functions for which these objects exist. As a first example of this notion of translator, we classify the analogues of the classical Grim Reapers within this context.

Submitted
Nov 26, 2022
Last revised
Sep 29, 2025 (v2)
Subjects
math.DG
Journal
Asian J. Math. vol 29 (2025), pp. 711-732
DOI
10.4310/AJM.260122040802

Comments: 27 pages. To appear in the Asian Journal of Mathematics

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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)5Martínez-Tri 2021 Translators to Higher Order Mean Curvature Flows in ℝⁿ×ℝ and ℍⁿ×ℝ (Ronaldo F. de Lima and Giuseppe Pipoli, 2022)8de 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)4de Lima 2023 The class of grim reapers in ℍ²×ℝ (Antonio Bueno and Rafael López, 2024)3Bueno 2024 Invariant λ-translators in Lorentz-Minkowski space (Antonio Bueno and Irene Ortiz, 2024)1Bueno 2024 Translators of the Mean Curvature Flow in Hyperbolic Einstein's Static Universe (Miguel Ortega and Buse Yalçın, 2024)2Ortega 2024 Translators of the mean curvature flow in the special linear group SL(2,ℝ) (Rafael López and Marian Ioan Munteanu, 2024)7López 2024 Characterizations of a Lorentzian Manifold with a semi-symmetric metric connection (Uday Chand De, Krishnendu De and Sinem Güler, 2024)6De 2024
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  1. Invariant -translators in Lorentz-Minkowski space

    Antonio Bueno and Irene OrtizWider flows

    Topics Physics, Self-shrinkers & translators · Shared terms “translating solitons”, “vector field”, “timelike”

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

    Miguel Ortega and Buse YalçınWider flows

    Topics Self-shrinkers & translators · Same author Miguel Ortega

  3. The class of grim reapers in

    Antonio Bueno and Rafael LópezWider flows

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

    Topics Self-shrinkers & translators · Shared terms “translating solitons”, “parameter family”, “distinguished”

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

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

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

  6. Characterizations of a Lorentzian Manifold with a semi-symmetric metric connection

    Uday Chand De, Krishnendu De and Sinem Güler

    Topics Physics · Shared terms “Lorentzian manifold”, “Robertson Walker”, “vector field”

  7. Translators of the mean curvature flow in the special linear group

    Rafael López and Marian Ioan MunteanuWider flows

    Topics Self-shrinkers & translators · Shared terms “vector field”, “Killing”

  8. Translators to Higher Order Mean Curvature Flows in and

    Ronaldo F. de Lima and Giuseppe PipoliWider flows

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

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Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

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.