Paper · Jun 19, 2019

Invariant translators of the Solvable group

Giuseppe Pipoli

math.DGWider flowsarXiv:1906.08077v2
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Abstract

We classify the translators to the mean curvature flow in the three-dimensional solvable group that are invariant under the action of a one-parameter group of isometries of the ambient space. In particular we show that admits graphical translators defined on a half-plane, in contrast with a rigidity result of Shahriyari for translators in the Euclidean space. Moreover we exhibit some non-existence results.

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Submitted
Jun 19, 2019
Last revised
Jul 17, 2019 (v2)
Subjects
math.DG

Comments: 18 pages, 7 figures, some typos have been corrected in the second version

Mean curvature flowSelf-shrinkers & translators3-manifoldsHomogeneous spacesStability & uniqueness

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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 Translators for Mean Curvature Flow (David Hoffman, Tom Ilmanen, Francisco Martin et al., 2018)7Hoffman 2018 Invariant translators of the Heisenberg group (Giuseppe Pipoli, 2018)1Pipoli 2018 Graphical translators for anisotropic and crystalline mean curvature flow (Annalisa Cesaroni, Heiko Kroener and Matteo Novaga, 2021)8Cesaroni 2021 Translators to Higher Order Mean Curvature Flows in ℝⁿ×ℝ and ℍⁿ×ℝ (Ronaldo F. de Lima and Giuseppe Pipoli, 2022)2de Lima 2022 The mean curvature flow on solvmanifolds (Romina M. Arroyo, Gabriela P. Ovando, Raquel Perales et al., 2023)6Arroyo 2023 Maximum Principles and Consequences for γ-translators in ℝⁿ⁺¹ (José Torres Santaella, 2023)4Santaella 2023 Translators invariant under hyperpolar actions (Tomoki Fujii and Naoyuki Koike, 2023)3Fujii 2023 Invariant λ-translators for the Gauss curvature flow in Euclidean space (Muhittin Evren Aydin and Rafael López, 2025)5Aydin 2025
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  1. Invariant translators of the Heisenberg group

    Giuseppe PipoliWider flows

    Topics 3-manifolds, Self-shrinkers & translators · Same author Giuseppe Pipoli · Shared terms “invariant translators”, “parameter group”, “contrast”

  2. Translators to Higher Order Mean Curvature Flows in and

    Ronaldo F. de Lima and Giuseppe PipoliWider flows

    Topics Self-shrinkers & translators, Stability & uniqueness · Same author Giuseppe Pipoli · Shared terms “Euclidean space”

  3. Translators invariant under hyperpolar actions

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  4. Maximum Principles and Consequences for -translators in

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    Topics Homogeneous spaces, Self-shrinkers & translators, Stability & uniqueness · Shared terms “translator defined”

  5. Invariant -translators for the Gauss curvature flow in Euclidean space

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    Topics Self-shrinkers & translators · Shared terms “parameter group”, “Euclidean space”

  6. The mean curvature flow on solvmanifolds

    Romina M. Arroyo, Gabriela P. Ovando, Raquel Perales et al.Wider flows

    Topics 3-manifolds, Homogeneous spaces, Self-shrinkers & translators · Shared terms “Euclidean space”

  7. Graphical Translators for Mean Curvature Flow

    David Hoffman, Tom Ilmanen, Francisco Martin et al.Wider flows

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

  8. Graphical translators for anisotropic and crystalline mean curvature flow

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

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) 3-manifolds Dimension three is where Ricci flow has done its most complete work. It proves geometrization, describes the flow's long-time behaviour, and now… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) 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)

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