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

John Hughes

TL;DR

The paper establishes the existence of Ricci flow with surgery in four dimensions that changes local topology from $\mathbb{CP}^2\setminus\{\mathrm{pt}\}$ to $\mathbb{R}^4$, with the post-surgery flow converging to Taub-NUT on $\mathbb{R}^4$ in infinite time. The approach relies on a $U(2)$-symmetric reduction, a spectral decomposition of the weighted Lichnerowicz Laplacian, and a Ważewski box argument adapted to a noncompact setting, yielding a finite-time singularity modeled on the FIK shrinker. The analysis shows curvature and geometric quantities remain controlled away from the singular bolt, preserves mass, and allows surgery to produce a flow on $\mathbb{R}^4$ that converges to Taub-NUT, with the limit determined by pre-surgery asymptotics. The results connect noncompact asymptotics, symmetry-reduced spectral theory, and explicit soliton models to demonstrate a robust 4D Ricci flow with topology change and long-term convergence to a hyperkähler metric.

Abstract

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

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

TL;DR

The paper establishes the existence of Ricci flow with surgery in four dimensions that changes local topology from to , with the post-surgery flow converging to Taub-NUT on in infinite time. The approach relies on a -symmetric reduction, a spectral decomposition of the weighted Lichnerowicz Laplacian, and a Ważewski box argument adapted to a noncompact setting, yielding a finite-time singularity modeled on the FIK shrinker. The analysis shows curvature and geometric quantities remain controlled away from the singular bolt, preserves mass, and allows surgery to produce a flow on that converges to Taub-NUT, with the limit determined by pre-surgery asymptotics. The results connect noncompact asymptotics, symmetry-reduced spectral theory, and explicit soliton models to demonstrate a robust 4D Ricci flow with topology change and long-term convergence to a hyperkähler metric.

Abstract

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

Paper Structure

This paper contains 29 sections, 35 theorems, 164 equations.

Table of Contents

  1. Introduction
  2. The Taub-NUT, Taub-Bolt and FIK metrics
  3. Nuts and bolts
  4. Four dimensional cohomogeneity one $SU(2)$-symmetric metrics
  5. Boundary conditions
  6. $\mathbb{R}^4$
  7. $\mathbb{R}^4$
  8. $\mathbb{CP}^2\setminus\{\mathrm{pt}\}$
  9. Three families of four dimensional cohomogeneity one $U(2)$-symmetric metrics
  10. Taub-NUT
  11. Taub-Bolt
  12. FIK
  13. $\Delta_{\overline{g},f}+2\overline{Rm}$ and its spectrum
  14. $U(2)$-symmetric basis of $L^{2,U(2)}_f$
  15. Ricci flow in $G_{AF, \mathbb{R}^4}$, $G_{AF,\mathbb{CP}^2\setminus\{\mathrm{pt}\}}$
  16. ...and 14 more sections

Key Result

Theorem 1.1

There is a $U(2)$-symmetric Ricci flow $G(t)$ with surgery such that

Theorems & Definitions (64)

  • Theorem 1.1
  • Theorem 2.1
  • proof
  • Definition 2.2
  • Theorem 2.3: Stol1, Theorem 2.19
  • Proposition 2.1
  • Theorem 2.4
  • Remark 2.5
  • Theorem 3.1
  • Proposition 3.1
  • ...and 54 more