Paper · Aug 10, 2020

Convergence of Ricci flow solutions to Taub-NUT

Francesco Di Giovanni

math.DGarXiv:2008.03969v2
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Abstract

We study the Ricci flow starting at an SU(2) cohomogeneity-1 metric on with monotone warping coefficients and whose restriction to any hypersphere is a Berger metric. If has bounded Hopf-fiber, curvature controlled by the size of the orbits and opens faster than a paraboloid in the directions orthogonal to the Hopf-fiber, then the flow converges to the Taub-NUT metric in the Cheeger-Gromov sense in infinite time. We also classify the long-time behaviour when is asymptotically flat. In order to identify infinite-time singularity models we obtain a uniqueness result for .

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Submitted
Aug 10, 2020
Last revised
Feb 17, 2021 (v2)
Subjects
math.DG, math.AP
DOI
10.1080/03605302.2021.1883651

Comments: 49 pages, final version. Accepted in Commun. Partial. Differ. Equ

Ricci flowSingularitiesHomogeneous spacesStability & uniquenessBoundaries & non-compactLong-time existence & convergence

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 Ricci flow of warped Berger metrics on ℝ⁴ (Francesco Di Giovanni, 2019)1Di Giovanni 2019 Rotationally symmetric Ricci flow on ℝⁿ⁺¹ (Francesco Di Giovanni, 2019)3Di Giovanni 2019 Singularity formation of complete Ricci flow solutions (Timothy Carson, James Isenberg, Dan Knopf et al., 2020)6Carson 2020 Infinite-time incompleteness of noncompact Yamabe flow (Jin Takahashi and Hikaru Yamamoto, 2021)7Takahashi 2021 On the Isometry Group of Immortal Homogeneous Ricci Flows (Roberto Araujo, 2023)5Araujo 2023 Long-time behavior of awesome homogeneous Ricci flows (Roberto Araujo, 2023)4Araujo 2023 Long-time Ricci flow existence and topological rigidity from manifolds with pinched scale-invariant integral curvature (Albert Chau and Adam Martens, 2024)8Chau 2024 Towards a Taub-Bolt to Taub-NUT via Ricci flow with surgery (John Hughes, 2025)2Hughes 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Ricci flow of warped Berger metrics on

    Francesco Di Giovanni

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

  2. Towards a Taub-Bolt to Taub-NUT via Ricci flow with surgery

    John Hughes

    Topics Singularities · Shared terms “Taub NUT”, “NUT metric”, “infinite time”

  3. Rotationally symmetric Ricci flow on

    Francesco Di Giovanni

    Topics Long-time existence & convergence, Boundaries & non-compact, Singularities · Same author Francesco Di Giovanni · Shared terms “hyperspheres”, “asymptotically flat”

  4. Long-time behavior of awesome homogeneous Ricci flows

    Roberto Araujo

    Topics Homogeneous spaces, Long-time existence & convergence, Boundaries & non-compact

  5. On the Isometry Group of Immortal Homogeneous Ricci Flows

    Roberto Araujo

    Topics Homogeneous spaces, Long-time existence & convergence, Boundaries & non-compact

  6. Singularity formation of complete Ricci flow solutions

    Timothy Carson, James Isenberg, Dan Knopf et al.

    Topics Boundaries & non-compact, Singularities, Stability & uniqueness · Shared terms “singularity models”

  7. Infinite-time incompleteness of noncompact Yamabe flow

    Jin Takahashi and Hikaru YamamotoWider flows

    Topics Long-time existence & convergence, Boundaries & non-compact, Singularities · Shared terms “infinite time”

  8. Long-time Ricci flow existence and topological rigidity from manifolds with pinched scale-invariant integral curvature

    Albert Chau and Adam Martens

    Topics Long-time existence & convergence, Boundaries & non-compact, Stability & uniqueness · Shared terms “asymptotically flat”, “flow starting”, “flow converges”

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) 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) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005)

In the guidebook

  1. 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.

  2. 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.

  3. 11A.13
    Homogeneous FlowsBook 11A · The Ricci Flow: Existence and Maximum Principles

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