Paper · May 3, 2023

The mean curvature flow on solvmanifolds

Romina M. Arroyo, Gabriela P. Ovando, Raquel Perales, Mariel Sáez

math.DGWider flowsarXiv:2305.02378v2
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Abstract

This work is a survey of the most relevant background material to motivate and understand the construction and classification of translating solutions to mean curvature flow on a family of solvmanifolds. We introduce the mean curvature flow and some known results in the field. In particular we explore the notion of translating solution in the Euclidean space and extensions into other Riemannian manifolds. We also include a discussion on solvmanifolds and some elements of its geometry that are relevant to our work. We finish by posing the equations that describe translating solutions to mean curvature flow on our family of 3-dimensional solvmanifolds with some additional assumptions. This project emerged at the "Latin American and Caribbean Workshop on Mathematics and Gender" held at Casa Matemática Oaxaca (CMO) from May 15-20, 2022.

Comments: In v2 we clarified that this manuscript is a survey paper and in particular added section 4

Mean curvature flowSelf-shrinkers & translators3-manifoldsHomogeneous spaces

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 Rotationally symmetric translating solutions to extrinsic geometric flows (Sathyanarayanan Rengaswami, 2021)3Rengaswami 2021 Inhomogeneous deformations of Einstein solvmanifolds (Adam Thompson, 2023)7Thompson 2023 Translators invariant under hyperpolar actions (Tomoki Fujii and Naoyuki Koike, 2023)6Fujii 2023 Uniqueness of Semigraphical Translators (Francisco Martín, Mariel Sáez and Raphael Tsiamis, 2023)4Martín 2023 Classification of Semigraphical Translators (Francisco Martín, Mariel Sáez, Raphael Tsiamis et al., 2024)2Martín 2024 Grim Raindrop: A Translating Solution to Curve Diffusion Flow (W. Jacob Ogden and Micah Warren, 2024)8Ogden 2024 Existence proofs for rotationally symmetric translating solutions to mean curvature flow (Hakar Raji and Oliver C. Schnürer, 2025)5Raji 2025 The mean curvature flow of subgroups on Lie groups of dimension three (Romina M. Arroyo, Gabriela P. Ovando and Mariel Sáez, 2025)1Arroyo 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. The mean curvature flow of subgroups on Lie groups of dimension three

    Romina M. Arroyo, Gabriela P. Ovando and Mariel SáezWider flows PDF

    Topics 3-manifolds, Homogeneous spaces · Same authors Romina M. Arroyo, Gabriela P. Ovando, Mariel Sáez

  2. Classification of Semigraphical Translators

    Francisco Martín, Mariel Sáez, Raphael Tsiamis et al.Wider flows PDF

    Topics Self-shrinkers & translators · Same author Mariel Sáez

  3. Rotationally symmetric translating solutions to extrinsic geometric flows

    Sathyanarayanan RengaswamiWider flows PDF

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

  4. Uniqueness of Semigraphical Translators

    Francisco Martín, Mariel Sáez and Raphael TsiamisWider flows PDF

    Topics Self-shrinkers & translators · Same author Mariel Sáez

  5. Existence proofs for rotationally symmetric translating solutions to mean curvature flow

    Hakar Raji and Oliver C. SchnürerWider flows PDF

    Topics Self-shrinkers & translators · Shared terms “translating solution”, “Euclidean space”

  6. Translators invariant under hyperpolar actions

    Tomoki Fujii and Naoyuki KoikeWider flows PDF

    Topics Self-shrinkers & translators, Homogeneous spaces

  7. Inhomogeneous deformations of Einstein solvmanifolds

    Adam Thompson PDF

    Topics Homogeneous spaces · Shared terms “solvmanifolds”

  8. Grim Raindrop: A Translating Solution to Curve Diffusion Flow

    W. Jacob Ogden and Micah WarrenWider flows PDF

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

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) 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)

In the guidebook

  1. 10A.1
    A Zoo of Three-ManifoldsBook 10A · Three-Manifolds and Geometrization

    The 3-sphere, lens spaces, the Poincaré sphere, the Hopf fibration and the shape of space.

  2. 10A.2
    Finite Rotation Groups and Spherical Space FormsBook 10A · Three-Manifolds and Geometrization

    Klein's classification by counting poles, the binary groups, and every quotient of the round 3-sphere.

  3. 10A.4
    Homotopy Groups, Degree and the Hurewicz TheoremBook 10A · Three-Manifolds and Geometrization

    Spheres mapped into manifolds: the algebraic topology that finite extinction runs on.