Paper · Oct 25, 2016

Ancient Solutions on Bundles with Non-abelian Structural Group

Peng Lu, Y. K. Wang

math.DGarXiv:1610.07706
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Abstract

We generalize the ancient solutions of the Ricci flow on certain principal bundles over compact quaternionic Kähler manifolds constructed by Bakas, Kong, and Ni to certain fibre bundles over a product of two compact quaternionic Kähler manifolds. The ancient solutions are of Type I, -noncollapsed, and have positive Ricci curvature. Analogous solutions on even-dimensional non-Kählerian bundle spaces are also constructed.

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Submitted
Oct 25, 2016
Subjects
math.DG

Comments: 33 pages

Ricci flowAncient solutionsSingularities(Non)collapsingKähler geometry

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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 Ricci Flow Solutions on Bundles (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)2Lu 2017 Ancient solutions for Andrews' hypersurface flow (Peng Lu and Jiuru Zhou, 2018)6Lu 2018 Uniqueness of ancient compact non-collapsed solutions to the 3-dimensional Ricci flow (Panagiota Daskalopoulos and Natasa Sesum, 2019)8Daskalopoulo 2019 Uniqueness of compact ancient solutions to three-dimensional Ricci flow (Simon Brendle, Panagiota Daskalopoulos and Natasa Sesum, 2020)5Brendle 2020 Ancient solutions to the Kähler Ricci flow (Yu Li, 2020)3Li 2020 Uniqueness of compact ancient solutions to the higher dimensional Ricci flow (Simon Brendle, Panagiota Daskalopoulos, Keaton Naff et al., 2021)4Brendle 2021 On the fundamental group of non-collapsed ancient Ricci flows (Richard H. Bamler, 2021)7Bamler 2021
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Ancient Ricci Flow Solutions on Bundles

    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. 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

  3. Ancient solutions to the Kähler Ricci flow

    Yu Li

    Topics (Non)collapsing, Ancient solutions, Kähler geometry · Shared terms “noncollapsed”

  4. Uniqueness of compact ancient solutions to the higher dimensional Ricci flow

    Simon Brendle, Panagiota Daskalopoulos, Keaton Naff et al.

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

  5. Uniqueness of compact ancient solutions to three-dimensional Ricci flow

    Simon Brendle, Panagiota Daskalopoulos and Natasa Sesum

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

  6. Ancient solutions for Andrews' hypersurface flow

    Peng Lu and Jiuru Zhou

    Topics Ancient solutions · Same author Peng Lu

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

    Richard H. Bamler

    Topics (Non)collapsing, Ancient solutions, Singularities

  8. Uniqueness of ancient compact non-collapsed solutions to the 3-dimensional Ricci flow

    Panagiota Daskalopoulos and Natasa Sesum

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

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) 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) (Non)collapsing At scales where curvature is controlled, balls in a Ricci flow cannot be arbitrarily thin: Perelman's κ-noncollapsing theorem. Start with: Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds (Cheeger et al., 1982)

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.