Paper · Sep 27, 2025

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

John Hughes

math.DGarXiv:2509.23257v2
PDF Abstract page

As on arXiv

Abstract

This paper shows for the first time the existence of a Ricci flow with surgery with local topology change \mathbbCP^2\setminus\{ pt\} \rightarrow \mathbbR^4. The post surgery flow converges to the Taub-NUT metric on \mathbbR^4 in infinite time.

Comments: Corrected some typos occurring version 1

Ricci flowSingularitiesSurgery

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)7Kleiner 2006 Singular Ricci flows I (Bruce Kleiner and John Lott, 2014)8Kleiner 2014 Canonical surgeries in rotationally invariant Ricci flow (Timothy Buttsworth, Maximilien Hallgren and Yongjia Zhang, 2022)6Buttsworth 2022 Cobordism, Singularities and the Ricci Flow Conjecture (David Martín Velázquez, Davide De Biasio and Dieter Lust, 2022)4Velázquez 2022 L^2-instability of the Taub-Bolt metric under the Ricci flow (John Hughes, 2024)2Hughes 2024 Passing through nondegenerate singularities in mean curvature flows (Ao Sun, Zhihan Wang and Jinxin Xue, 2025)3Sun 2025 An ancient Ricci flow emerging from Taub-Bolt (John Hughes, 2025)1Hughes 2025 Mean convex flows with surgery (Gábor Székelyhidi, 2026)5Székelyhidi 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. An ancient Ricci flow emerging from Taub-Bolt

    John Hughes PDF

    Topics Ricci flow · Same author John Hughes · Shared terms “Taub Bolt”

  2. L^2-instability of the Taub-Bolt metric under the Ricci flow

    John Hughes PDF

    Topics Singularities · Same author John Hughes · Shared terms “Taub Bolt”

  3. Passing through nondegenerate singularities in mean curvature flows

    Ao Sun, Zhihan Wang and Jinxin XueWider flows PDF

    Topics Surgery, Singularities · Shared terms “topology change”

  4. Cobordism, Singularities and the Ricci Flow Conjecture

    David Martín Velázquez, Davide De Biasio and Dieter Lust PDF

    Topics Surgery, Singularities

  5. Mean convex flows with surgery

    Gábor SzékelyhidiWider flows PDF

    Topics Surgery, Singularities

  6. Canonical surgeries in rotationally invariant Ricci flow

    Timothy Buttsworth, Maximilien Hallgren and Yongjia Zhang PDF

    Topics Surgery, Singularities

  7. Notes on Perelman's papers

    Bruce Kleiner and John LottThe canon PDF

    Topics Surgery

  8. Singular Ricci flows I

    Bruce Kleiner and John LottThe canon PDF

    Topics Surgery

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)

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).