Paper · Apr 3, 2019

Ricci flow of warped Berger metrics on

Francesco Di Giovanni

math.DGarXiv:1904.02236v4
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Abstract

We study the Ricci flow on starting at an SU(2)-cohomogeneity 1 metric whose restriction to any hypersphere is a Berger metric. We prove that if has no necks and is bounded by a cylinder, then the solution develops a global Type-II singularity and converges to the Bryant soliton when suitably dilated at the origin. This is the first example in dimension of a non-rotationally symmetric Type-II flow converging to a rotationally symmetric singularity model. Next, we show that if instead has no necks, its curvature decays and the Hopf fibers are not collapsed, then the solution is immortal. Finally, we prove that if the flow is Type-I, then there exist minimal 3-spheres for times close to the maximal time.

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Submitted
Apr 3, 2019
Last revised
Feb 17, 2021 (v4)
Subjects
math.DG
Journal
Calc. Var. 59, 162 (2020)
DOI
10.1007/s00526-020-01823-4

Comments: 40 pages, final version. Accepted in Calc. Var

Ricci flowAncient solutionsSingularitiesHomogeneous spacesLong-time existence & convergenceRotational 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 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 Rotational symmetry of ancient solutions to the Ricci flow in dimension 3 – The compact case (S. Brendle, 2019)4Brendle 2019 Rotationally symmetric Ricci flow on ℝⁿ⁺¹ (Francesco Di Giovanni, 2019)1Di Giovanni 2019 Classification of 3-dimensional complete rectifiable steady and expanding gradient Ricci solitons (Shun Maeta, 2019)7Maeta 2019 Ricci Flow Singularity for Triaxial Bianchi IX Metric (M. Syafiq Johar, 2019)6Johar 2019 On the Ricci Flow of Homogeneous Metrics on Spheres (Sammy Sbiti, 2020)3Sbiti 2020 Convergence of Ricci flow solutions to Taub-NUT (Francesco Di Giovanni, 2020)2Di Giovanni 2020 Non-homothetic convex ancient solutions for flows by high powers of curvature (Susanna Risa and Carlo Sinestrari, 2022)8Risa 2022 Rotationally symmetric Ricci Flow on ℝⁿ⁺¹ (Ming Hsiao, 2025)5Hsiao 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Rotationally symmetric Ricci flow on

    Francesco Di Giovanni

    Topics Rotational symmetry & warped products, Ancient solutions, Long-time existence & convergence · Same author Francesco Di Giovanni · Shared terms “rotationally symmetric”, “solution develops”, “Bryant soliton”

  2. Convergence of Ricci flow solutions to Taub-NUT

    Francesco Di Giovanni

    Topics Homogeneous spaces, Long-time existence & convergence, Singularities · Same author Francesco Di Giovanni · Shared terms “Hopf fiber”, “Berger metric”, “hyperspheres”

  3. On the Ricci Flow of Homogeneous Metrics on Spheres

    Sammy Sbiti

    Topics Ancient solutions, Homogeneous spaces · Shared terms “Berger metrics”, “collapsed”

  4. Rotational symmetry of ancient solutions to the Ricci flow in dimension – The compact case

    S. Brendle

    Topics Rotational symmetry & warped products, Ancient solutions · Shared terms “Bryant soliton”, “rotationally symmetric”

  5. Rotationally symmetric Ricci Flow on

    Ming Hsiao

    Topics Rotational symmetry & warped products, Singularities · Shared terms “rotationally symmetric”, “hyperspheres”

  6. Ricci Flow Singularity for Triaxial Bianchi IX Metric

    M. Syafiq Johar

    Topics Rotational symmetry & warped products, Homogeneous spaces, Singularities · Shared terms “rotationally symmetric”

  7. Classification of 3-dimensional complete rectifiable steady and expanding gradient Ricci solitons

    Shun Maeta

    Topics Rotational symmetry & warped products, Ancient solutions · Shared terms “Bryant soliton”, “rotationally symmetric”

  8. Non-homothetic convex ancient solutions for flows by high powers of curvature

    Susanna Risa and Carlo SinestrariWider flows

    Topics Rotational symmetry & warped products, Ancient solutions, Homogeneous spaces · Shared terms “rotationally symmetric”

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

    Necks and caps, compactness, and universal estimates.

  2. 11B.4
    Blow-Ups, Blow-Downs and Splitting at InfinityBook 11B · Solitons, Compactness and Singularities

    Limits of rescaled manifolds and flows: why blow-up limits are nonnegatively curved, what blow-downs see, and how lines appear at infinity.

  3. 11B.6
    Rotationally Symmetric Flows and the Standard SolutionBook 11B · Solitons, Compactness and Singularities

    The neckpinch in equations: warped products under the flow, what the maximum principle forces, and the standard solution that surgery glues in.