Paper · May 23, 2025

The mean curvature flow of subgroups on Lie groups of dimension three

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

math.DGWider flowsarXiv:2505.17892v2
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Abstract

In this work we study the existence of solutions to the Mean Curvature Flow for which the initial condition has the structure of a two-dimensional Lie subgroup within a Lie group of dimension three. We consider Lie groups with a fixed left-invariant metric and first observe that if the Lie group is unimodular, then every Lie subgroup is a minimal surface (hence a trivial solution). For this reason we focus on non-unimodular Lie groups, finding the evolution of every Lie subgroup of dimension 2 (within a 3 dimensional Lie group). These evolutions are self-similar for abelian subgroups (i.e. evolve by isometries), but not self-similar in the other cases.

Mean curvature flow3-manifoldsSurfacesHomogeneous spacesMinimal & CMC surfaces

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  1. The mean curvature flow on solvmanifolds

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

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

  2. Classification of Minimal Surfaces and Solitons to the Mean Curvature Flow in as Translation Surfaces

    Tarcios Andrey Ferreira and João Paulo dos SantosWider flows PDF

    Topics Minimal & CMC surfaces, Surfaces, Homogeneous spaces · Shared terms “Lie groups”

  3. Ricci Solitons of Left-Invariant Metrics on Lie Groups with Derived Algebra of Dimension at Most Two

    Hamid Reza Salimi Moghaddam PDF

    Topics Surfaces, Homogeneous spaces · Shared terms “Lie groups”, “left invariant”

  4. Codimension one Ricci soliton subgroups of nilpotent Iwasawa groups

    Victor Sanmartin-Lopez PDF

    Topics Homogeneous spaces · Shared terms “Lie subgroup”, “Lie groups”, “isometry”

  5. Ricci Solitons on the Lie Group and Applications

    Mohammed El Amine Mekki and Ahmed Mohammed Cherif PDF

    Topics 3-manifolds, Homogeneous spaces · Shared terms “Lie groups”, “left invariant”

  6. Ricci solitons as submanifolds of complex hyperbolic spaces

    Ángel Cidre-Díaz and Víctor Sanmartín-López PDF

    Topics Homogeneous spaces · Shared terms “Lie subgroup”

  7. Algebraic and Symmetrized Affine star-Ricci Solitons for the Bott Connection on Three-Dimensional Lorentzian Lie Groups

    Yuxin Ding PDF

    Topics Surfaces, 3-manifolds, Homogeneous spaces · Shared terms “Lie groups”, “unimodular”, “left invariant”

  8. On the long-time limit of the mean curvature flow in closed manifolds

    Alexander Mramor and Ao SunWider flows PDF

    Topics Minimal & CMC surfaces, 3-manifolds

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) 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) Surfaces On surfaces Ricci flow is conformal and scalar. It gives a heat-flow proof of the uniformization theorem: every closed surface converges to constant… Start with: The Ricci flow on surfaces (Hamilton, 1988) 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. 5A.5
    Uniformization and the Two-Dimensional Ricci FlowBook 5A · Complex Analysis and Conformal Geometry

    Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.

  2. 11A.12
    Ricci Flow on SurfacesBook 11A · The Ricci Flow: Existence and Maximum Principles

    Uniformization by Ricci flow, and Hamilton’s entropy.

  3. 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.