Paper · Jan 3, 2017

Immortal homogeneous Ricci flows

Christoph Böhm, Ramiro A. Lafuente

math.DGarXiv:1701.00628v4
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Abstract

We show that for an immortal homogeneous Ricci flow solution any sequence of parabolic blow-downs subconverges to a homogeneous expanding Ricci soliton. This is established by constructing a new Lyapunov function based on curvature estimates which come from real geometric invariant theory.

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Submitted
Jan 3, 2017
Last revised
Nov 17, 2017 (v4)
Subjects
math.DG
DOI
10.1007/s00222-017-0771-z

Comments: Final version, to appear in Invent. Math

Ricci flowRicci solitonsHomogeneous spacesLong-time existence & convergence

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)2Böhm 2016 The Ricci flow on solvmanifolds of real type (Christoph Böhm and Ramiro A. Lafuente, 2017)3Böhm 2017 On the Isometry Group of Immortal Homogeneous Ricci Flows (Roberto Araujo, 2023)1Araujo 2023 Long-time behavior of awesome homogeneous Ricci flows (Roberto Araujo, 2023)8Araujo 2023 The homogeneous generalized Ricci flow (Elia Fusi, Ramiro A. Lafuente and James Stanfield, 2024)5Fusi 2024 Expanding Ricci solitons and Higgs bundles (Ramiro A. Lafuente and Adam Thompson, 2025)4Lafuente 2025 On the long time behavior of ancient homogeneous Ricci flows (Anusha M. Krishnan and Francesco Pediconi, 2026)7Krishnan 2026 Ricci flows and bracket flows on twisted principal bundles (Henry Wood, 2026)6Wood 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. On the Isometry Group of Immortal Homogeneous Ricci Flows

    Roberto Araujo

    Topics Homogeneous spaces, Long-time existence & convergence · Shared terms “immortal homogeneous”

  2. Optimal curvature estimates for homogeneous Ricci flows

    Christoph Böhm, Ramiro Lafuente and Miles Simon

    Topics Homogeneous spaces, Long-time existence & convergence · Same authors Christoph Böhm, Ramiro A. Lafuente · Shared terms “curvature estimates”, “immortal”

  3. The Ricci flow on solvmanifolds of real type

    Christoph Böhm and Ramiro A. Lafuente

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

  4. Expanding Ricci solitons and Higgs bundles

    Ramiro A. Lafuente and Adam Thompson

    Topics Homogeneous spaces, Long-time existence & convergence, Ricci solitons · Same author Ramiro A. Lafuente · Shared terms “expanding Ricci”

  5. The homogeneous generalized Ricci flow

    Elia Fusi, Ramiro A. Lafuente and James Stanfield

    Topics Homogeneous spaces, Long-time existence & convergence, Ricci solitons · Same author Ramiro A. Lafuente

  6. Ricci flows and bracket flows on twisted principal bundles

    Henry Wood

    Topics Homogeneous spaces, Long-time existence & convergence, Ricci solitons · Shared terms “homogeneous expanding”, “immortal”, “expanding Ricci”

  7. On the long time behavior of ancient homogeneous Ricci flows

    Anusha M. Krishnan and Francesco Pediconi

    Topics Homogeneous spaces, Long-time existence & convergence, Ricci solitons · Shared terms “sequence”

  8. Long-time behavior of awesome homogeneous Ricci flows

    Roberto Araujo

    Topics Homogeneous spaces, Long-time existence & convergence

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.

Foundational paper in the Homogeneous spaces guide: Böhm–Lafuente: immortal homogeneous Ricci flows.

Foundational paper in the Long-time existence & convergence guide: Böhm–Lafuente: immortal homogeneous Ricci flows.

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) Ricci solitons Metrics that change under Ricci flow only by scaling and diffeomorphisms: shrinking, steady or expanding solitons. 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) Long-time existence & convergence A geometric flow either runs forever or stops because something blows up. Extension criteria say which quantity must blow up at a finite singular… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982)

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. 11B.1
    Ricci SolitonsBook 11B · Solitons, Compactness and Singularities

    Self-similar solutions: the Gaussian, cylinders, the cigar and the Bryant soliton.

  3. 12B.2
    The Asymptotic Soliton and Three-Dimensional ShrinkersBook 12B · κ-Solutions and Surgery

    Perelman's Proposition 11.2 proved with Morgan and Tian: the reduced-length bounds on κ-solutions, the limit of rescalings, the equality case, and the classification of shrinking solitons in dimensions two and three.