Paper · Feb 28, 2018

Painlevé analysis of Ricci solitons over warped products

Alejandro Betancourt de la Parra

math.DGarXiv:1802.10227
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Abstract

We carry out a Painlevé analysis to find the cases where the cohomogeneity one steady Ricci soliton equation can be integrable. We concentrate on two classes of solitons: warped products and complex line bundles over a Fano Kähler Einstein base. For warped products, the analysis singles out the case with one factor where the dimension of the hypersurface is a perfect square, with the particularly distinguished. The case with two factors each of dimension is also singled out by the analysis. In the case of complex line bundles, a 1-parameter family is singled out for every even dimension.

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Submitted
Feb 28, 2018
Subjects
math.DG
Ricci solitonsEinstein metricsKähler geometryHomogeneous spacesRotational symmetry & warped productsKähler–Einstein & K-stability

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 Destabilising compact warped product Einstein manifolds (Wafaa Batat, Stuart James Hall and Thomas Murphy, 2016)7Batat 2016 Classification of base of warped product almost Ricci solitons (José N. V. Gomes and Manoel V. M. Neto, 2017)3Gomes 2017 A family of non-collapsed steady Ricci solitons in even dimensions greater or equal to four (Alexander Appleton, 2017)2Appleton 2017 Rigidity of Homogeneous Gradient Soliton Metrics and Related Equations (Peter Petersen and William Wylie, 2020)8Petersen 2020 On noncompact warped product Ricci solitons (Valter Borges, 2022)6Borges 2022 Backward propagation of warped product structures and asymptotically conical shrinkers (Brett Kotschwar, 2023)4Kotschwar 2023 Infinitely many non-collapsed steady Ricci solitons on complex line bundles (Hanci Chi, 2024)1Chi 2024 Ricci Flow on CP1-bundles over a Product of Kähler-Einstein Manifolds (Frederick Tsz-Ho Fong and Hung Tran, 2026)5Fong 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Infinitely many non-collapsed steady Ricci solitons on complex line bundles

    Hanci Chi

    Topics Kähler–Einstein & K-stability, Einstein metrics, Kähler geometry · Shared terms “complex line”, “line bundle”, “steady Ricci”

  2. A family of non-collapsed steady Ricci solitons in even dimensions greater or equal to four

    Alexander Appleton

    Topics Kähler–Einstein & K-stability, Einstein metrics, Kähler geometry · Shared terms “complex line”, “line bundle”, “steady Ricci”

  3. Classification of base of warped product almost Ricci solitons

    José N. V. Gomes and Manoel V. M. Neto

    Topics Rotational symmetry & warped products, Einstein metrics, Ricci solitons

  4. Backward propagation of warped product structures and asymptotically conical shrinkers

    Brett Kotschwar

    Topics Rotational symmetry & warped products, Einstein metrics, Ricci solitons

  5. Ricci Flow on CP1-bundles over a Product of Kähler-Einstein Manifolds

    Frederick Tsz-Ho Fong and Hung Tran

    Topics Kähler–Einstein & K-stability, Einstein metrics, Kähler geometry

  6. On noncompact warped product Ricci solitons

    Valter Borges

    Topics Rotational symmetry & warped products, Einstein metrics, Ricci solitons

  7. Destabilising compact warped product Einstein manifolds

    Wafaa Batat, Stuart James Hall and Thomas Murphy

    Topics Rotational symmetry & warped products, Einstein metrics, Ricci solitons

  8. Rigidity of Homogeneous Gradient Soliton Metrics and Related Equations

    Peter Petersen and William Wylie

    Topics Rotational symmetry & warped products, Einstein metrics, Homogeneous spaces

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) Kähler geometry Complex manifolds with compatible Riemannian metrics whose Kähler form is closed. Here Ricci curvature is the complex Hessian of a log-determinant… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) 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.