Paper · May 23, 2018

Ricci flow from some spaces with asymptotically cylindrical singularities

Timothy Carson

math.DGarXiv:1805.09401
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Abstract

We prove the existence of Ricci flow starting from a class of metrics with unbounded curvature, which are doubly-warped products over an interval with a spherical factor pinched off at an end. These provide a forward evolution from some known and conjectured finite-time local singularities of Ricci flow, generalizing previous examples. The class also includes metrics with non-compact singular ends which become instantaneously compact. Furthermore, we prove local stability of the forward evolution, which allows us to glue it to other manifolds and create a forward evolution from spaces which are not globally warped products.

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Submitted
May 23, 2018
Subjects
math.DG
Ricci flowSingularitiesStability & uniquenessBoundaries & non-compactRotational 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 A Pohozaev Identity on Warped Product Solitons (Timothy Carson, 2016)2Carson 2016 Ricci Flow recovering from pinched discs (Timothy Carson, 2017)1Carson 2017 Curvature Blow-up in Doubly-warped Product Metrics Evolving by Ricci Flow (Maxwell Stolarski, 2019)4Stolarski 2019 Singularity formation of complete Ricci flow solutions (Timothy Carson, James Isenberg, Dan Knopf et al., 2020)3Carson 2020 Convergence stability for Ricci flow on manifolds with bounded geometry (Eric Bahuaud, Christine Guenther, James Isenberg et al., 2023)8Bahuaud 2023 Backward propagation of warped product structures and asymptotically conical shrinkers (Brett Kotschwar, 2023)6Kotschwar 2023 Local singularities of compact multiply warped Ricci flow solutions (James Isenberg, Dan Knopf, Zilu Ma et al., 2025)5Isenberg 2025 Gradient estimates for a parabolic partial differential equation under the Ricci-Bourguignon flow (José N. V. Gomes, Willian I. Tokura and Hikaru Yamamoto, 2026)7Gomes 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Ricci Flow recovering from pinched discs

    Timothy Carson

    Topics Rotational symmetry & warped products, Singularities · Same author Timothy Carson · Shared terms “forward evolution”, “doubly warped”, “flow starting”

  2. A Pohozaev Identity on Warped Product Solitons

    Timothy Carson

    Topics Rotational symmetry & warped products, Singularities · Same author Timothy Carson

  3. Singularity formation of complete Ricci flow solutions

    Timothy Carson, James Isenberg, Dan Knopf et al.

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

  4. Curvature Blow-up in Doubly-warped Product Metrics Evolving by Ricci Flow

    Maxwell Stolarski

    Topics Rotational symmetry & warped products, Singularities · Shared terms “doubly warped”

  5. Local singularities of compact multiply warped Ricci flow solutions

    James Isenberg, Dan Knopf, Zilu Ma et al.

    Topics Rotational symmetry & warped products, Boundaries & non-compact, Singularities · Shared terms “local singularity”

  6. Backward propagation of warped product structures and asymptotically conical shrinkers

    Brett Kotschwar

    Topics Rotational symmetry & warped products, Boundaries & non-compact

  7. Gradient estimates for a parabolic partial differential equation under the Ricci-Bourguignon flow

    José N. V. Gomes, Willian I. Tokura and Hikaru Yamamoto

    Topics Rotational symmetry & warped products, Boundaries & non-compact

  8. Convergence stability for Ricci flow on manifolds with bounded geometry

    Eric Bahuaud, Christine Guenther, James Isenberg et al.

    Topics Rotational symmetry & warped products, Boundaries & non-compact, Singularities · Shared terms “finite time”

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.