Paper · Jan 16, 2020

Singularity formation of complete Ricci flow solutions

Timothy Carson, James Isenberg, Dan Knopf, Natasa Sesum

math.DGarXiv:2001.06098
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Abstract

We study singularity formation of complete Ricci flow solutions, motivated by two applications: (a) improving the understanding of the behavior of the essential blowup sequences of Enders-Muller-Topping on noncompact manifolds, and (b) obtaining further evidence in favor of the conjectured stability of generalized cylinders as Ricci flow singularity models.

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Submitted
Jan 16, 2020
Subjects
math.DG
Ricci flowSingularitiesStability & uniquenessBoundaries & non-compact

Its neighbourhood

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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 Non-Kahler Ricci flow singularities modeled on Kahler-Ricci solitons (James Isenberg, Dan Knopf and Natasa Sesum, 2017)1Isenberg 2017 Dynamic instability of ℂP}^N under Ricci flow (Dan Knopf and Natasa Sesum, 2017)6Knopf 2017 Ricci flow from some spaces with asymptotically cylindrical singularities (Timothy Carson, 2018)7Carson 2018 A numerical stability analysis of mean curvature flow of noncompact hypersurfaces with Type-II curvature blowup (David Garfinkle, James Isenberg, Dan Knopf et al., 2021)4Garfinkle 2021 A numerical stability analysis of mean curvature flow of noncompact hypersurfaces with Type-II curvature blowup: II (David Garfinkle, James Isenberg, Dan Knopf et al., 2022)5Garfinkle 2022 Asymptotic behavior of unstable perturbations of the Fubini-Study metric in Ricci flow (David Garfinkle, James Isenberg, Dan Knopf et al., 2024)3Garfinkle 2024 Local singularities of compact multiply warped Ricci flow solutions (James Isenberg, Dan Knopf, Zilu Ma et al., 2025)2Isenberg 2025 Infinite-dimensional dynamical instabilities of noncompact stationary Ricci flow solutions (Sigurd B. Angenent and Dan Knopf, 2025)8Angenent 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Non-Kahler Ricci flow singularities modeled on Kahler-Ricci solitons

    James Isenberg, Dan Knopf and Natasa Sesum

    Topics Singularities, Stability & uniqueness · Same authors James Isenberg, Dan Knopf, Natasa Sesum · Shared terms “conjectured stability”, “singularity models”, “evidence”

  2. Local singularities of compact multiply warped Ricci flow solutions

    James Isenberg, Dan Knopf, Zilu Ma et al.

    Topics Boundaries & non-compact, Singularities · Same authors James Isenberg, Dan Knopf, Natasa Sesum · Shared terms “generalized cylinder”, “formation”, “singularity models”

  3. Asymptotic behavior of unstable perturbations of the Fubini-Study metric in Ricci flow

    David Garfinkle, James Isenberg, Dan Knopf et al.

    Topics Singularities, Stability & uniqueness · Same authors James Isenberg, Dan Knopf

  4. A numerical stability analysis of mean curvature flow of noncompact hypersurfaces with Type-II curvature blowup

    David Garfinkle, James Isenberg, Dan Knopf et al.Wider flows

    Topics Boundaries & non-compact, Singularities, Stability & uniqueness · Same authors James Isenberg, Dan Knopf

  5. A numerical stability analysis of mean curvature flow of noncompact hypersurfaces with Type-II curvature blowup: II

    David Garfinkle, James Isenberg, Dan Knopf et al.Wider flows

    Topics Boundaries & non-compact, Singularities, Stability & uniqueness · Same authors James Isenberg, Dan Knopf · Shared terms “evidence”

  6. Dynamic instability of under Ricci flow

    Dan Knopf and Natasa Sesum

    Topics Stability & uniqueness · Same authors Dan Knopf, Natasa Sesum

  7. Ricci flow from some spaces with asymptotically cylindrical singularities

    Timothy Carson

    Topics Boundaries & non-compact, Singularities, Stability & uniqueness · Same author Timothy Carson · Shared terms “conjectured”

  8. Infinite-dimensional dynamical instabilities of noncompact stationary Ricci flow solutions

    Sigurd B. Angenent and Dan Knopf

    Topics Boundaries & non-compact, Stability & uniqueness · Same author Dan Knopf

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) 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) Boundaries & non-compact Ricci flow beyond closed manifolds: complete non-compact spaces, manifolds with boundary, and asymptotically flat or hyperbolic ends. Start with: Deforming the metric on complete Riemannian manifolds (Shi, 1989)

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. 6A.13
    Self-Similar Variables and MonotonicityBook 6A · The Heat Equation and Its Relatives

    Changing variables so that self-similar solutions stand still: convergence to the Gaussian, Giga–Kohn blow-up, Gaussian-weighted monotonicity formulas, frequency functions, and the blow-up logic of Huisken, Struwe and Perelman.