Paper · Aug 30, 2023

Translators invariant under hyperpolar actions

Tomoki Fujii, Naoyuki Koike

math.DGWider flowsarXiv:2308.15790v2
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Abstract

In this paper, we consider translators (for the mean curvature flow) given by a graph of a function on a symmetric space of compact type which is invariant under a hyperpolar action on . First, in the case of , , or , we classify the shapes of translators in given by the graphs of functions on which are invariant under the isotropy action . Next, in the case where is of higher rank, we investigate translators in given by the graphs of functions on which are invariant under a hyperpolar action of cohomogeneity two.

Comments: 19 pages

Mean curvature flowSelf-shrinkers & translatorsHomogeneous spaces

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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 Graphical translating solitons for the mean curvature flow and isoparametric functions (Tomoki Fujii, 2021)3Fujii 2021 Translators of the Gauss curvature flow (Muhittin Evren Aydin and Rafael López, 2022)8Aydin 2022 Graphical translating solitons for the inverse mean curvature flow and isoparametric functions (Tomoki Fujii, 2022)7Fujii 2022 Maximum Principles and Consequences for γ-translators in ℝⁿ⁺¹ (José Torres Santaella, 2023)6Santaella 2023 Isotropy invariant graphical mean curvature flows in warped products (Naotoshi Fujihara and Naoyuki Koike, 2024)4Fujihara 2024 Mean curvature flow for principal orbits of Hermann actions on rank two symmetric spaces (Naoyuki Koike and Sakura Nakaoka, 2025)1Koike 2025 Invariant λ-translators in the Heisenberg group (Antonio Bueno, 2026)5Bueno 2026 Cohomogeneity one Hermann actions and backward mean curvature flows (Kurando Baba and Naoyuki Koike, 2026)2Baba 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
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    Naoyuki Koike and Sakura NakaokaWider flows PDF

    Topics Homogeneous spaces · Same author Naoyuki Koike · Shared terms “symmetric space”, “compact type”, “cohomogeneity”

  2. Cohomogeneity one Hermann actions and backward mean curvature flows

    Kurando Baba and Naoyuki KoikeWider flows PDF

    Topics Homogeneous spaces · Same author Naoyuki Koike · Shared terms “compact type”, “symmetric space”, “cohomogeneity”

  3. Graphical translating solitons for the mean curvature flow and isoparametric functions

    Tomoki FujiiWider flows PDF

    Topics Self-shrinkers & translators · Same author Tomoki Fujii

  4. Isotropy invariant graphical mean curvature flows in warped products

    Naotoshi Fujihara and Naoyuki KoikeWider flows PDF

    Topics Mean curvature flow · Same author Naoyuki Koike · Shared terms “compact type”, “symmetric space”, “isotropy”

  5. Invariant -translators in the Heisenberg group

    Antonio BuenoWider flows PDF

    Topics Self-shrinkers & translators · Shared terms “translators invariant”

  6. Maximum Principles and Consequences for -translators in

    José Torres SantaellaWider flows PDF

    Topics Self-shrinkers & translators, Homogeneous spaces

  7. Graphical translating solitons for the inverse mean curvature flow and isoparametric functions

    Tomoki FujiiWider flows PDF

    Topics Self-shrinkers & translators · Same author Tomoki Fujii

  8. Translators of the Gauss curvature flow

    Muhittin Evren Aydin and Rafael LópezWider flows PDF

    Topics Self-shrinkers & translators · Shared terms “translators invariant”

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) 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. 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.