Paper · Jun 12, 2025

Expanding Ricci solitons and Higgs bundles

Ramiro A. Lafuente, Adam Thompson

math.DGarXiv:2506.11362v3
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Abstract

Motivated by the long-time behavior of Ricci flows that collapse with bounded curvature, we study expanding Ricci solitons with nilpotent symmetry on vector bundles over a closed manifold. We prove that, under mild assumptions that are satisfied by Ricci flow limits, the equations dimension-reduce to the so-called twisted harmonic-Einstein equations. When the base is a surface, we establish a correspondence between solutions of the latter and a class of G-Higgs bundles. This allows us to produce infinite families of new examples that are not locally homogeneous, and in particular to obtain a complete description in dimension 4. We also show that all our examples admit Einstein one-dimensional extensions.

Comments: 33 pages; v2: introduced affine expanding solitons, improved presentation and main results; v3: fixed cross-referencing typos from v2 due to a LaTeX bug

Ricci flowRicci solitons4-manifoldsHomogeneous spacesLong-time existence & convergence

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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 Four-dimensional generalized Ricci flows with nilpotent symmetry (Steven Gindi and Jeffrey Streets, 2021)3Gindi 2021 Inhomogeneous deformations of Einstein solvmanifolds (Adam Thompson, 2023)4Thompson 2023 New expanding Ricci solitons starting in dimension four (Jan Nienhaus and Matthias Wink, 2023)5Nienhaus 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)2Fusi 2024 Ricci Flows with Nilpotent Symmetry and Zero Bundle Curvature (Steven Gindi, 2024)1Gindi 2024 Pseudo-Riemannian Algebraic Ricci Solitons on Four-Dimensional Lie Groups (Youssef Ayad, 2026)6Ayad 2026 A note on the classification of four-dimensional gradient steady and expanding Ricci solitons (Huai-Dong Cao and Junming Xie, 2026)7Cao 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 PDF

    Topics 4-manifolds, Homogeneous spaces, Ricci solitons · Shared terms “nilpotent symmetry”, “expanding Ricci”

  2. The homogeneous generalized Ricci flow

    Elia Fusi, Ramiro A. Lafuente and James Stanfield PDF

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

  3. Four-dimensional generalized Ricci flows with nilpotent symmetry

    Steven Gindi and Jeffrey Streets PDF

    Topics 4-manifolds, Homogeneous spaces, Long-time existence & convergence · Shared terms “nilpotent symmetry”

  4. Inhomogeneous deformations of Einstein solvmanifolds

    Adam Thompson PDF

    Topics Homogeneous spaces, Ricci solitons · Same author Adam Thompson · Shared terms “Einstein”

  5. New expanding Ricci solitons starting in dimension four

    Jan Nienhaus and Matthias Wink PDF

    Topics 4-manifolds, Ricci solitons · Shared terms “expanding Ricci”, “vector bundles”, “Einstein”

  6. Pseudo-Riemannian Algebraic Ricci Solitons on Four-Dimensional Lie Groups

    Youssef Ayad PDF

    Topics 4-manifolds, Homogeneous spaces, Ricci solitons · Shared terms “complete description”

  7. A note on the classification of four-dimensional gradient steady and expanding Ricci solitons

    Huai-Dong Cao and Junming Xie PDF

    Topics 4-manifolds, Ricci solitons · Shared terms “expanding Ricci”

  8. Long-time behavior of awesome homogeneous Ricci flows

    Roberto Araujo PDF

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

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) 4-manifolds Four is the first dimension where Ricci flow's singularities are not fully understood. It is also the dimension of the smooth Poincaré conjecture… Start with: The topology of four-dimensional manifolds (Freedman, 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. 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.