Paper · Apr 21, 2017

Ricci Flow recovering from pinched discs

Timothy Carson

math.DGarXiv:1704.06385
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As on arXiv

Abstract

We construct smooth solutions to Ricci flow starting from a class of singular metrics and give asymptotics for the forward evolution. The singular metrics heal with a set of points (of codimension at least three) coming out of the singular point. We conjecture that these metrics arise as final-time limits of Ricci flow encountering a Type-I singularity modeled on . This gives a picture of Ricci flow through a singularity, in which a neighborhood of the manifold changes topology from to (through the cone over .) We work in the class of doubly-warped product metrics. We also briefly discuss some possible smooth and non-smooth forward evolutions from other singular initial data.

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Submitted
Apr 21, 2017
Subjects
math.DG, math.AP
Ricci flowSingularitiesWeak & singular flowsRotational 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 from some spaces with asymptotically cylindrical singularities (Timothy Carson, 2018)1Carson 2018 Curvature Blow-up in Doubly-warped Product Metrics Evolving by Ricci Flow (Maxwell Stolarski, 2019)3Stolarski 2019 On Weak Super Ricci Flow through Neckpinch (Sajjad Lakzian and Michael Munn, 2020)7Lakzian 2020 Rotational symmetry of solutions of mean curvature flow coming out of a double cone (Letian Chen, 2021)5Chen 2021 Rotational symmetry of solutions of mean curvature flow coming out of a double cone II (Letian Chen, 2022)4Chen 2022 Exponential Asymptotics of Ricci Flow Neckpinch (Hamidreza Mahmoudian, 2023)6Mahmoudian 2023 Rotationally symmetric Ricci Flow on ℝⁿ⁺¹ (Ming Hsiao, 2025)8Hsiao 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 from some spaces with asymptotically cylindrical singularities

    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 · Shared terms “product metrics”

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

    Maxwell Stolarski

    Topics Rotational symmetry & warped products, Singularities · Shared terms “doubly warped”, “product metrics”, “singularity modeled”

  4. Rotational symmetry of solutions of mean curvature flow coming out of a double cone II

    Letian ChenWider flows

    Topics Rotational symmetry & warped products, Weak & singular flows, Singularities

  5. Rotational symmetry of solutions of mean curvature flow coming out of a double cone

    Letian ChenWider flows

    Topics Rotational symmetry & warped products, Weak & singular flows · Shared terms “smooth solution”

  6. Exponential Asymptotics of Ricci Flow Neckpinch

    Hamidreza Mahmoudian

    Topics Rotational symmetry & warped products, Singularities · Shared terms “asymptotics”, “initial data”

  7. On Weak Super Ricci Flow through Neckpinch

    Sajjad Lakzian and Michael Munn

    Topics Weak & singular flows, Singularities · Shared terms “forward evolution”, “smooth forward”, “final time”

  8. Rotationally symmetric Ricci Flow on

    Ming Hsiao

    Topics Rotational symmetry & warped products, Weak & singular flows, Singularities

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) Weak & singular flows Running a flow when the data isn't smooth or singularities form: Ricci flows from rough initial data, through singularities and on metric measure… Start with: Deformation of C⁰ Riemannian metrics in the direction of their Ricci curvature (Simon, 2002) Rotational symmetry & warped products For rotationally symmetric metrics and warped products, Ricci flow reduces to a parabolic equation in one space variable. Start with: The ricci flow on radially symmetric R3 (Ivey, 1994)

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. 12C.9
    After PerelmanBook 12C · Extinction and the Proof

    Singular flows, the Smale conjecture, and Bamler’s theory.