Paper · Nov 2, 2026

Ricci flows and bracket flows on twisted principal bundles

Henry Wood

math.DGarXiv:2611.02477
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As on arXiv

Abstract

We show that Ricci flows of invariant metrics on twisted principal bundles with nilpotent fibres over are immortal and type III. After normalisation and lift to the universal cover, a sequence of metrics converges in the Cheeger–Gromov topology to a homogeneous expanding Ricci soliton. The proof extends the bracket flow technique from homogeneous Ricci flows to invariant Ricci flows on twisted principal bundles.

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Submitted
Nov 2, 2026
Subjects
math.DG
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 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 On the Isometry Group of Immortal Homogeneous Ricci Flows (Roberto Araujo, 2023)4Araujo 2023 Long-time behavior of awesome homogeneous Ricci flows (Roberto Araujo, 2023)2Araujo 2023 The homogeneous generalized Ricci flow (Elia Fusi, Ramiro A. Lafuente and James Stanfield, 2024)3Fusi 2024 Ricci Flows with Nilpotent Symmetry and Zero Bundle Curvature (Steven Gindi, 2024)1Gindi 2024 3-symmetric spaces, Ricci solitons, and homogeneous structures (Thomas Murphy and Paul-Andi Nagy, 2025)6Murphy 2025 Expanding Ricci solitons and Higgs bundles (Ramiro A. Lafuente and Adam Thompson, 2025)5Lafuente 2025 On the long time behavior of ancient homogeneous Ricci flows (Anusha M. Krishnan and Francesco Pediconi, 2026)7Krishnan 2026 Long-Time Behavior of the α-Ricci Flow on the 2-Torus (Yifan Guo and Xiaokang Wang, 2026)8Guo 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Ricci Flows with Nilpotent Symmetry and Zero Bundle Curvature

    Steven Gindi

    Topics Homogeneous spaces, Ricci solitons · Shared terms “principal bundles”, “invariant Ricci”, “nilpotent”

  2. Long-time behavior of awesome homogeneous Ricci flows

    Roberto Araujo

    Topics Homogeneous spaces, Long-time existence & convergence · Shared terms “universal cover”

  3. The homogeneous generalized Ricci flow

    Elia Fusi, Ramiro A. Lafuente and James Stanfield

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

  4. On the Isometry Group of Immortal Homogeneous Ricci Flows

    Roberto Araujo

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

  5. Expanding Ricci solitons and Higgs bundles

    Ramiro A. Lafuente and Adam Thompson

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

  6. 3-symmetric spaces, Ricci solitons, and homogeneous structures

    Thomas Murphy and Paul-Andi Nagy

    Topics Homogeneous spaces, Long-time existence & convergence, Ricci solitons · Shared terms “Type III”, “expanding Ricci”, “nilpotent”

  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 “invariant metrics”, “sequence”

  8. Long-Time Behavior of the -Ricci Flow on the 2-Torus

    Yifan Guo and Xiaokang Wang

    Topics Long-time existence & convergence, Ricci solitons · Shared terms “universal cover”, “Type III”, “Gromov”

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) 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.