Paper · Jan 10, 2024

Three-dimensional positively curved generalized Ricci solitons with SO(3)-symmetries

Fabio Podestà, Alberto Raffero

math.DGarXiv:2401.05028v3
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Abstract

We prove the existence of a one-parameter family of pairwise non-isometric, complete, positively curved, steady generalized Ricci solitons of gradient type on that are invariant under the natural cohomogeneity one action of SO(3). In the context of generalized Ricci flow, this result represents the analogue of Bryant's construction of the complete rotationally invariant steady soliton for the Ricci flow.

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Submitted
Jan 10, 2024
Last revised
Jul 7, 2025 (v3)
Subjects
math.DG
Journal
Advances in Mathematics, 479, 110426, 2025
DOI
10.1016/j.aim.2025.110426
Ricci flowGeneralized & G₂ flowsRicci solitons3-manifoldsHomogeneous spacesRotational symmetry & warped products

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 Bismut Ricci flat manifolds with symmetries (Fabio Podestà and Alberto Raffero, 2022)1Podestà 2022 On the dynamics of positively curved metrics on SU(3)/T² under the homogeneous Ricci flow (Leonardo F. Cavenaghi, Lino Grama and Ricardo M. Martins, 2023)7Cavenaghi 2023 The homogeneous generalized Ricci flow (Elia Fusi, Ramiro A. Lafuente and James Stanfield, 2024)6Fusi 2024 Dynamical stability of Pluriclosed and Generalized Ricci solitons (Kuan-Hui Lee, 2025)8Lee 2025 Cylindrical generalized Ricci solitons in three dimensions (Miguel Pino Carmona, 2026)2Carmona 2026 Uniqueness of positively curved ancient Ricci flows on surfaces with boundary (Kyeongho Bang, Eric Chen and Wenkui Du, 2026)3Bang 2026 Solvable Algebraic Generalized Ricci Solitons in low dimensions (Lorenzo Mucciante, 2026)4Mucciante 2026 Harmonic torsion of algebraic generalized Ricci solitons and the structure of solvable extensions (Huibin Chen, Zhiqi Chen and Fuhai Zhu, 2026)5Chen 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Bismut Ricci flat manifolds with symmetries

    Fabio Podestà and Alberto Raffero

    Topics Generalized & G₂ flows, Homogeneous spaces · Same authors Fabio Podestà, Alberto Raffero

  2. Cylindrical generalized Ricci solitons in three dimensions

    Miguel Pino Carmona

    Topics Generalized & G₂ flows, Rotational symmetry & warped products, 3-manifolds · Shared terms “parameter family”

  3. Uniqueness of positively curved ancient Ricci flows on surfaces with boundary

    Kyeongho Bang, Eric Chen and Wenkui Du

    Topics Rotational symmetry & warped products · Shared terms “positively curved”, “rotationally”

  4. Solvable Algebraic Generalized Ricci Solitons in low dimensions

    Lorenzo Mucciante

    Topics Generalized & G₂ flows, 3-manifolds, Homogeneous spaces

  5. Harmonic torsion of algebraic generalized Ricci solitons and the structure of solvable extensions

    Huibin Chen, Zhiqi Chen and Fuhai Zhu

    Topics Generalized & G₂ flows, Homogeneous spaces, Ricci solitons

  6. The homogeneous generalized Ricci flow

    Elia Fusi, Ramiro A. Lafuente and James Stanfield

    Topics Generalized & G₂ flows, Homogeneous spaces, Ricci solitons

  7. On the dynamics of positively curved metrics on under the homogeneous Ricci flow

    Leonardo F. Cavenaghi, Lino Grama and Ricardo M. Martins

    Topics Homogeneous spaces · Shared terms “positively curved”

  8. Dynamical stability of Pluriclosed and Generalized Ricci solitons

    Kuan-Hui Lee

    Topics Generalized & G₂ flows, Ricci solitons · Shared terms “steady solitons”

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) Generalized & G₂ flows Ricci-type flows for geometry beyond Kähler metrics. Generalized Ricci flow couples a metric to a closed 3-form, pluriclosed and Chern–Ricci flows… Start with: A parabolic flow of pluriclosed metrics (Streets et al., 2009) 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) 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)

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  2. 10A.2
    Finite Rotation Groups and Spherical Space FormsBook 10A · Three-Manifolds and Geometrization

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  3. 10A.4
    Homotopy Groups, Degree and the Hurewicz TheoremBook 10A · Three-Manifolds and Geometrization

    Spheres mapped into manifolds: the algebraic topology that finite extinction runs on.