Paper · Jun 3, 2016

Ancient Ricci Flow Solutions on Bundles

Peng Lu, Y. K. Wang

math.DGarXiv:1606.01112
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Abstract

We generalize the circle bundle examples of ancient solutions of the Ricci flow discovered by Bakas, Kong, and Ni to a class of principal torus bundles over an arbitrary finite product of Fano Kähler-Einstein manifolds studied by Wang and Ziller in the context of Einstein geometry. As a result, continuous families of -collapsed and -noncollapsed ancient solutions of type I are obtained on circle bundles for all odd dimensions . In dimension such examples moreover exist on pairs of homeomorphic but not diffeomorphic manifolds. Continuous families of -collapsed ancient solutions of type I are also obtained on torus bundles for all dimensions .

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Submitted
Jun 3, 2016
Subjects
math.DG

Comments: 40 pages, no figure

Ricci flowAncient solutionsEinstein metricsSingularities(Non)collapsingKähler geometryKähler–Einstein & K-stability

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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 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 Ancient Solutions on Bundles with Non-abelian Structural Group (Peng Lu and Y. K. Wang, 2016)1Lu 2016 New proofs of Perelman's theorem on shrinking Breathers in Ricci flow (Peng Lu and Yu Zheng, 2017)6Lu 2017 A family of non-collapsed steady Ricci solitons in even dimensions greater or equal to four (Alexander Appleton, 2017)8Appleton 2017 Ricci Flow on Torus Bundles (Dmytro Yeroshkin, 2021)3Yeroshkin 2021 On the fundamental group of non-collapsed ancient Ricci flows (Richard H. Bamler, 2021)4Bamler 2021 Toral symmetries of collapsed ancient solutions to the homogeneous Ricci flow (Anusha M. Krishnan, Francesco Pediconi and Sammy Sbiti, 2023)2Krishnan 2023 Ricci Flow on CP1-bundles over a Product of Kähler-Einstein Manifolds (Frederick Tsz-Ho Fong and Hung Tran, 2026)5Fong 2026 Finite-Time Singularities of the Kähler–Ricci Flow on a ℂP^m-Bundle over a Product of Kähler–Einstein Manifolds (Yifan Xiao, 2026)7Xiao 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Ancient Solutions on Bundles with Non-abelian Structural Group

    Peng Lu and Y. K. Wang

    Topics (Non)collapsing, Ancient solutions, Kähler geometry · Same authors Peng Lu, Y. K. Wang · Shared terms “Bakas”, “Kong”, “noncollapsed”

  2. Toral symmetries of collapsed ancient solutions to the homogeneous Ricci flow

    Anusha M. Krishnan, Francesco Pediconi and Sammy Sbiti

    Topics (Non)collapsing, Einstein metrics, Ancient solutions · Shared terms “torus bundles”, “collapsed ancient”, “principal torus”

  3. Ricci Flow on Torus Bundles

    Dmytro Yeroshkin

    Topics Kähler–Einstein & K-stability, Einstein metrics, Kähler geometry · Shared terms “torus bundles”, “circle bundle”

  4. On the fundamental group of non-collapsed ancient Ricci flows

    Richard H. Bamler

    Topics (Non)collapsing, Ancient solutions, Singularities · Shared terms “collapsed ancient”

  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 · Shared terms “Wang”

  6. New proofs of Perelman's theorem on shrinking Breathers in Ricci flow

    Peng Lu and Yu Zheng

    Topics (Non)collapsing, Ancient solutions, Singularities · Same author Peng Lu · Shared terms “collapsed”

  7. Finite-Time Singularities of the Kähler–Ricci Flow on a -Bundle over a Product of Kähler–Einstein Manifolds

    Yifan Xiao

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

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

    Alexander Appleton

    Topics Kähler–Einstein & K-stability, (Non)collapsing, Einstein metrics · Shared terms “collapsed”

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) Ancient solutions Solutions that exist for all negative times. Every blow-up of a singularity is one, so classifying ancient solutions means classifying the shapes… Start with: The entropy formula for the Ricci flow and its geometric applications (Perelman, 2002) 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) Singularities On a closed manifold the flow can stop in finite time: curvature blows up, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993)

In the guidebook

  1. 12B.4
    Compactness and Universal Estimates for κ-SolutionsBook 12B · κ-Solutions and Surgery

    Asymptotic volume zero, curvature controlled by volume, Perelman's Theorem 11.7, a universal κ, neck-like behaviour at infinity, and the constants that canonical neighbourhoods inherit.

  2. 12B.3
    The Structure of κ-SolutionsBook 12B · κ-Solutions and Surgery

    Necks and caps, compactness, and universal estimates.

  3. 9B.3
    Collapsing and NoncollapsingBook 9B · Comparison, Convergence and Heat on Manifolds

    Injectivity radius bounds from volume, collapsing examples, and the definition Perelman needs.