Paper · Nov 13, 2020

Translating Solitons in a Lorentzian Setting, Submersions and Cohomogeneity One Actions

Marie-Amelie Lawn, Miguel Ortega

math.DGWider flowsarXiv:2011.06915v2
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Abstract

We study new examples of translating solitons of the mean curvature flow, especially in Minkowski space. We consider for this purpose manifolds admitting submersions and cohomegeneity one actions by isometries on suitable open subsets. This general setting also covers the classical Euclidean examples. As an application, we completely classify timelike, invariant translating solitons by rotations and boosts in Minkowski space.

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Submitted
Nov 13, 2020
Last revised
Apr 9, 2021 (v2)
Subjects
math.DG, math.AP

Comments: 24 pages, 3 figures

Mean curvature flowSelf-shrinkers & translatorsHomogeneous spacesPhysics

Its neighbourhood

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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 2016, 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 A Zoo of Translating Solitons on a Parallel Light-like Direction in Minkowski 3-Space (Erdem Kocakuşaklı and Miguel Ortega, 2020)2Kocakuşaklı 2020 The prescribed curvature problem for entire hypersurfaces in Minkowski space (Changyu Ren, Zhizhang Wang and Ling Xiao, 2020)5Ren 2020 SO(2) symmetry of the translating solitons of the mean curvature flow in ℝ⁴ (Jingze Zhu, 2020)6Zhu 2020 Entire σₖ curvature flow in Minkowski space (Zhizhang Wang and Ling Xiao, 2022)8Wang 2022 New Examples of Translating Solitons in Generalised Robertson-Walker Geometries (Diego Artacho, Marie-Amélie Lawn and Miguel Ortega, 2022)1Artacho 2022 Translators invariant under hyperpolar actions (Tomoki Fujii and Naoyuki Koike, 2023)4Fujii 2023 Invariant λ-translators in Lorentz-Minkowski space (Antonio Bueno and Irene Ortiz, 2024)3Bueno 2024 Translating solutions and the entire Hessian curvature flow in Minkowski space (Qu Changzheng, Wang Zhizhang and Wo Weifeng, 2024)7Changzheng 2024
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. New Examples of Translating Solitons in Generalised Robertson-Walker Geometries

    Diego Artacho, Marie-Amélie Lawn and Miguel OrtegaWider flows

    Topics Physics, Self-shrinkers & translators · Same authors Marie-Amelie Lawn, Miguel Ortega · Shared terms “translating solitons”, “Lorentzian”, “timelike”

  2. A Zoo of Translating Solitons on a Parallel Light-like Direction in Minkowski 3-Space

    Erdem Kocakuşaklı and Miguel OrtegaWider flows

    Topics Self-shrinkers & translators · Same author Miguel Ortega · Shared terms “translating solitons”, “Minkowski”

  3. Invariant -translators in Lorentz-Minkowski space

    Antonio Bueno and Irene OrtizWider flows

    Topics Physics, Self-shrinkers & translators · Shared terms “Minkowski space”, “translating solitons”, “rotation”

  4. Translators invariant under hyperpolar actions

    Tomoki Fujii and Naoyuki KoikeWider flows

    Topics Homogeneous spaces, Self-shrinkers & translators · Shared terms “cohomogeneity”

  5. The prescribed curvature problem for entire hypersurfaces in Minkowski space

    Changyu Ren, Zhizhang Wang and Ling XiaoWider flows

    Topics Physics, Self-shrinkers & translators · Shared terms “Minkowski space”, “translating solitons”

  6. symmetry of the translating solitons of the mean curvature flow in

    Jingze ZhuWider flows

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

  7. Translating solutions and the entire Hessian curvature flow in Minkowski space

    Qu Changzheng, Wang Zhizhang and Wo WeifengWider flows

    Topics Physics, Self-shrinkers & translators · Shared terms “Minkowski space”

  8. Entire curvature flow in Minkowski space

    Zhizhang Wang and Ling XiaoWider flows

    Topics Physics, Self-shrinkers & translators · Shared terms “Minkowski space”

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 in the L² sense. 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) 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) Physics Ricci flow first appeared in physics, as the leading-order renormalization-group flow of the string sigma model (Friedan, 1980). Start with: Nonlinear Models in 2+ε Dimensions (Friedan, 1980)

In the guidebook

  1. 11A.13
    Homogeneous FlowsBook 11A · The Ricci Flow: Existence and Maximum Principles

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

  2. 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.