Paper · Jan 23, 2022

Canonical surgeries in rotationally invariant Ricci flow

Timothy Buttsworth, Maximilien Hallgren, Yongjia Zhang

math.DGarXiv:2201.09387v2
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Abstract

We construct a rotationally invariant Ricci flow through surgery starting at any closed rotationally invariant Riemannian manifold. We demonstrate that a sequence of such Ricci flows with surgery converges to a Ricci flow spacetime in the sense of [32]. Results of Bamler-Kleiner [8] and Haslhofer [29] then guarantee the uniqueness and stability of these spacetimes given initial data. We simplify aspects of this proof in our setting, and show that for rotationally invariant Ricci flows, the closeness of spacetimes can be measured by equivariant comparison maps. Finally we show that the blowup rate of the curvature near a singular time for these Ricci flows is bounded by the inverse of remaining time squared.

Comments: 48 pages

Ricci flowSingularitiesSurgeryStability & uniquenessRotational 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 2021, 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 Notes on Perelman's papers (Bruce Kleiner and John Lott, 2006)6Kleiner 2006 Singular Ricci flows I (Bruce Kleiner and John Lott, 2014)1Kleiner 2014 Hamilton-Ivey estimates for gradient Ricci solitons (Pak-Yeung Chan, Zilu Ma and Yongjia Zhang, 2021)5Chan 2021 Cobordism, Singularities and the Ricci Flow Conjecture (David Martín Velázquez, Davide De Biasio and Dieter Lust, 2022)4Velázquez 2022 Exponential Asymptotics of Ricci Flow Neckpinch (Hamidreza Mahmoudian, 2023)8Mahmoudian 2023 Towards a Taub-Bolt to Taub-NUT via Ricci flow with surgery (John Hughes, 2025)3Hughes 2025 Nondegenerate neck pinches along the mean curvature flow (Gábor Székelyhidi, 2026)7Székelyhidi 2026 Mean convex flows with surgery (Gábor Székelyhidi, 2026)2Székelyhidi 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Singular Ricci flows I

    Bruce Kleiner and John LottThe canon PDF

    Topics Surgery · Shared terms “surgery starting”, “flow spacetime”

  2. Mean convex flows with surgery

    Gábor SzékelyhidiWider flows PDF

    Topics Surgery, Singularities · Shared terms “surgery starting”, “Kleiner”, “Haslhofer”

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

    John Hughes PDF

    Topics Surgery, Singularities

  4. Cobordism, Singularities and the Ricci Flow Conjecture

    David Martín Velázquez, Davide De Biasio and Dieter Lust PDF

    Topics Surgery, Singularities

  5. Hamilton-Ivey estimates for gradient Ricci solitons

    Pak-Yeung Chan, Zilu Ma and Yongjia Zhang PDF

    Topics Rotational symmetry & warped products, Singularities · Same author Yongjia Zhang · Shared terms “rotationally”

  6. Notes on Perelman's papers

    Bruce Kleiner and John LottThe canon PDF

    Topics Surgery

  7. Nondegenerate neck pinches along the mean curvature flow

    Gábor SzékelyhidiWider flows PDF

    Topics Surgery, Singularities · Shared terms “singular time”

  8. Exponential Asymptotics of Ricci Flow Neckpinch

    Hamidreza Mahmoudian PDF

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

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 usually cannot run forever: curvature blows up in finite time, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993) Surgery Stop the flow just before a neck pinches, cut along the thin neck, glue standard caps onto the two ends, and restart. Start with: Four-manifolds with positive isotropic curvature (Hamilton, 1997) 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. 12B.5
    The Canonical Neighbourhood TheoremBook 12B · κ-Solutions and Surgery

    Every high-curvature point looks like a neck or a cap.

  2. 12B.7
    The Singular Time: Horns and Strong NecksBook 12B · κ-Solutions and Surgery

    The limit metric at a singular time, the set Ω where curvature stays bounded, ε-horns and their ends, and strong δ-necks deep in a horn, following Perelman II §3 and Morgan–Tian chapter 11.

  3. 12B.12
    Canonical Neighbourhoods After SurgeryBook 12B · κ-Solutions and Surgery

    The strong canonical neighbourhood assumption justified across surgeries: the blow-up argument, the three cases for a limit that meets a surgery, and the choice of δ (Perelman II §5; Morgan–Tian chapter 17).