Paper · Jul 13, 2017

The Ricci flow on solvmanifolds of real type

Christoph Böhm, Ramiro A. Lafuente

math.DGarXiv:1707.04186v2
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Abstract

We show that for any solvable Lie group of real type, any homogeneous Ricci flow solution converges in Cheeger-Gromov topology to a unique non-flat solvsoliton, which is independent of the initial left-invariant metric. As an application, we obtain results on the isometry groups of non-flat solvsoliton metrics and Einstein solvmanifolds.

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Submitted
Jul 13, 2017
Last revised
Aug 22, 2017 (v2)
Subjects
math.DG

Comments: 18 pages; v2: improved exposition, proofs of main results simplified, appendices are now contained in arXiv:1701.00628v3

Ricci flowHomogeneous 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 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 Optimal curvature estimates for homogeneous Ricci flows (Christoph Böhm, Ramiro Lafuente and Miles Simon, 2016)3Böhm 2016 Immortal homogeneous Ricci flows (Christoph Böhm and Ramiro A. Lafuente, 2017)1Böhm 2017 Maximal Symmetry and Unimodular Solvmanifolds (Michael Jablonski, 2018)2Jablonski 2018 Hermitian Curvature flow on unimodular Lie groups and static invariant metrics (Ramiro A. Lafuente, Mattia Pujia and Luigi Vezzoni, 2018)5Lafuente 2018 Einstein solvmanifolds as submanifolds of symmetric spaces (Michael Jablonski, 2018)6Jablonski 2018 Infinitesimal maximal symmetry and Ricci soliton solvmanifolds (Carolyn Gordon and Michael Jablonski, 2021)4Gordon 2021 Inhomogeneous deformations of Einstein solvmanifolds (Adam Thompson, 2023)8Thompson 2023 The homogeneous generalized Ricci flow (Elia Fusi, Ramiro A. Lafuente and James Stanfield, 2024)7Fusi 2024
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Immortal homogeneous Ricci flows

    Christoph Böhm and Ramiro A. Lafuente

    Topics Homogeneous spaces · Same authors Christoph Böhm, Ramiro A. Lafuente

  2. Maximal Symmetry and Unimodular Solvmanifolds

    Michael Jablonski

    Topics Homogeneous spaces · Shared terms “solvmanifolds”, “isometry group”, “solvable Lie”

  3. Optimal curvature estimates for homogeneous Ricci flows

    Christoph Böhm, Ramiro Lafuente and Miles Simon

    Topics Homogeneous spaces · Same authors Christoph Böhm, Ramiro A. Lafuente

  4. Infinitesimal maximal symmetry and Ricci soliton solvmanifolds

    Carolyn Gordon and Michael Jablonski

    Topics Homogeneous spaces · Shared terms “solvmanifolds”, “solvable Lie”, “isometry group”

  5. Hermitian Curvature flow on unimodular Lie groups and static invariant metrics

    Ramiro A. Lafuente, Mattia Pujia and Luigi Vezzoni

    Topics Homogeneous spaces · Same author Ramiro A. Lafuente · Shared terms “Lie groups”, “left invariant”, “Cheeger Gromov”

  6. Einstein solvmanifolds as submanifolds of symmetric spaces

    Michael Jablonski

    Topics Homogeneous spaces · Shared terms “solvmanifolds”, “solvable Lie”, “Lie groups”

  7. The homogeneous generalized Ricci flow

    Elia Fusi, Ramiro A. Lafuente and James Stanfield

    Topics Homogeneous spaces · Same author Ramiro A. Lafuente · Shared terms “solvmanifolds”, “Lie groups”

  8. Inhomogeneous deformations of Einstein solvmanifolds

    Adam Thompson

    Topics Homogeneous spaces · Shared terms “solvmanifolds”

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.

Ricci flow Hamilton's Ricci flow evolves a Riemannian metric by minus twice its Ricci curvature. It is a nonlinear heat equation for geometry: it spreads… 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. 11A.13
    Homogeneous FlowsBook 11A · The Ricci Flow: Existence and Maximum Principles

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

  2. 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.

  3. 10A.12
    Curve Shortening and Minimal DiscsBook 10A · Three-Manifolds and Geometrization

    How loops shrink under curvature, how the discs they bound lose area, and Perelman's extinction estimate.