Paper · Jun 29, 2017

Cohomogeneity one Ricci Solitons from Hopf Fibrations

Matthias Wink

math.DGarXiv:1706.09712v2
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Abstract

This paper studies cohomogeneity one Ricci solitons. If the isotropy representation of the principal orbit consists of two inequivalent -invariant irreducible summands, the existence of parameter families of non-homothetic complete steady and expanding Ricci solitons on non-trivial bundles is shown. These examples were detected numerically by Buzano-Dancer-Gallaugher-Wang. The analysis of the corresponding Ricci flat trajectories is used to reconstruct Einstein metrics of positive scalar curvature due to Böhm. The techniques also apply to -quasi-Einstein metrics.

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Submitted
Jun 29, 2017
Last revised
Oct 27, 2019 (v2)
Subjects
math.DG
Journal
Comm. Anal. Geom. 31 (2023), no. 3, 625-676
DOI
10.4310/CAG.2023.v31.n3.a4

Comments: 34 pages, asymptotics refined

Ricci solitonsEinstein metricsRicci-flat & Calabi–YauAlmost & η-Ricci solitonsScalar curvatureHomogeneous spacesNumerics & visualization

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 Complete Ricci solitons via estimates on the soliton potential (Matthias Wink, 2017)2Wink 2017 New expanding Ricci solitons starting in dimension four (Jan Nienhaus and Matthias Wink, 2023)1Nienhaus 2023 Non-Shrinking Ricci Solitons of cohomogeneity one from the quaternionic Hopf fibration (Hanci Chi, 2024)4Chi 2024 Cohomogeneity one 4-dimensional gradient Ricci solitons (Patrick Donovan, 2025)7Donovan 2025 A Characterization of Quasi-Einstein Metrics (Antonio Airton Freitas Filho, 2025)8Filho 2025 Cohomogeneity One Expanding Ricci Solitons and the Expander Degree (Abishek Rajan, 2025)3Rajan 2025 Quasi-Einstein Metrics and a curvature identity associated with the Ricci flow (Atreyee Bhattacharya and Sayoojya Prakash, 2025)6Bhattacharya 2025 Homogeneous spaces with two equivalent isotropy summands develop positive Ricci curvature under Ricci flow (Eric Cochran, Arseny Mingajev, Lawrence Mouillé et al., 2026)5Cochran 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. New expanding Ricci solitons starting in dimension four

    Jan Nienhaus and Matthias Wink

    Topics Ricci-flat & Calabi–Yau, Einstein metrics, Scalar curvature · Same author Matthias Wink · Shared terms “expanding Ricci”, “positive scalar”

  2. Complete Ricci solitons via estimates on the soliton potential

    Matthias Wink

    Topics Homogeneous spaces, Ricci solitons · Same author Matthias Wink · Shared terms “complete steady”, “cohomogeneity”, “Dancer”

  3. Cohomogeneity One Expanding Ricci Solitons and the Expander Degree

    Abishek Rajan

    Topics Homogeneous spaces, Ricci solitons · Shared terms “cohomogeneity”, “Buzano Dancer”, “Dancer Gallaugher”

  4. Non-Shrinking Ricci Solitons of cohomogeneity one from the quaternionic Hopf fibration

    Hanci Chi

    Topics Homogeneous spaces, Ricci solitons · Shared terms “Hopf fibration”, “cohomogeneity”, “parameter family”

  5. Homogeneous spaces with two equivalent isotropy summands develop positive Ricci curvature under Ricci flow

    Eric Cochran, Arseny Mingajev, Lawrence Mouillé et al.

    Topics Numerics & visualization, Homogeneous spaces · Shared terms “isotropy representation”, “irreducible”

  6. Quasi-Einstein Metrics and a curvature identity associated with the Ricci flow

    Atreyee Bhattacharya and Sayoojya Prakash

    Topics Almost & η-Ricci solitons, Einstein metrics, Ricci solitons · Shared terms “quasi Einstein”

  7. Cohomogeneity one 4-dimensional gradient Ricci solitons

    Patrick Donovan

    Topics Numerics & visualization, Homogeneous spaces, Ricci solitons · Shared terms “cohomogeneity”, “parameter family”

  8. A Characterization of Quasi-Einstein Metrics

    Antonio Airton Freitas Filho

    Topics Almost & η-Ricci solitons, Einstein metrics, Scalar curvature · Shared terms “quasi Einstein”

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 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) Einstein metrics Metrics whose Ricci curvature is a constant multiple of the metric. They are the fixed points of normalised Ricci flow, and the most natural metrics… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) Ricci-flat & Calabi–Yau Ricci-flat metrics do not move under Ricci flow. Every known compact simply connected example has special holonomy (Calabi–Yau, hyperkähler, G₂… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) Almost & η-Ricci solitons Variations on the Ricci soliton equation: let the constant λ become a function (almost solitons), add a term along a contact form (η-Ricci… Start with: Ricci almost solitons (Pigola et al., 2010)

In the guidebook

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    Computing Geometric FlowsBook 6D · Diffusions, Finance and Computation

    Numerical curve shortening and mean curvature flow, level sets and phase fields, discrete Ricci flow on surfaces and in conformal mapping, rotationally symmetric neckpinches, and what computation has taught about singularities.

  2. 11A.13
    Homogeneous FlowsBook 11A · The Ricci Flow: Existence and Maximum Principles

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

  3. 6C.4
    The Yamabe Problem and Critical ExponentsBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Conformal changes of scalar curvature, the Yamabe functional, Aubin's threshold and Schoen's solution, concentration compactness and bubbles, and why Perelman's log-Sobolev minimisers always exist.