Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Computer-assisted construction of $SU(2)$-invariant negative Einstein metrics

Qiu Shi Wang

TL;DR

The paper proves the existence of a 2-parameter family of complete negative Einstein metrics that are $SU(2)$-invariant and cohomogeneity-one on the complex line bundle $\mathcal{O}(-4)$, with metrics that are conformally compact and asymptotically hyperbolic yet generally non-Kähler and non-self-dual. The authors combine a local singular-orbit analysis, a fixed-point argument for a nearby true solution, and a global infinity extension via a conformally compact reformulation, all supported by rigorous computer-assisted numerics using Chebyshev approximations and interval arithmetic. The method builds on and extends ideas from Buttsworth–Hodgkinson, providing a robust, reproducible pipeline to certify new triaxial Einstein metrics and suggesting applicability to other cohomogeneity-one curvature problems. The results contribute explicit examples of nontrivial negative Einstein geometries with controlled asymptotics, enriching the landscape of known Bianchi IX solutions and offering a practical framework for future computer-assisted geometric analysis.

Abstract

We construct a 2-parameter family of new triaxial $SU(2)$-invariant complete negative Einstein metrics on the complex line bundle $\mathcal{O}(-4)$ over $\mathbb{C}P^1$. The metrics are conformally compact and generically neither Kähler nor self-dual. The proof involves using rigorous numerics to produce an approximate Einstein metric to high precision in a bounded region containing the singular orbit or "bolt", which is then perturbed to a genuine Einstein metric using fixed-point methods. At the boundary of this region, the latter metric is sufficiently close to hyperbolic space for us to show that it indeed extends to a complete, asymptotically hyperbolic Einstein metric.

Computer-assisted construction of $SU(2)$-invariant negative Einstein metrics

TL;DR

The paper proves the existence of a 2-parameter family of complete negative Einstein metrics that are -invariant and cohomogeneity-one on the complex line bundle , with metrics that are conformally compact and asymptotically hyperbolic yet generally non-Kähler and non-self-dual. The authors combine a local singular-orbit analysis, a fixed-point argument for a nearby true solution, and a global infinity extension via a conformally compact reformulation, all supported by rigorous computer-assisted numerics using Chebyshev approximations and interval arithmetic. The method builds on and extends ideas from Buttsworth–Hodgkinson, providing a robust, reproducible pipeline to certify new triaxial Einstein metrics and suggesting applicability to other cohomogeneity-one curvature problems. The results contribute explicit examples of nontrivial negative Einstein geometries with controlled asymptotics, enriching the landscape of known Bianchi IX solutions and offering a practical framework for future computer-assisted geometric analysis.

Abstract

We construct a 2-parameter family of new triaxial -invariant complete negative Einstein metrics on the complex line bundle over . The metrics are conformally compact and generically neither Kähler nor self-dual. The proof involves using rigorous numerics to produce an approximate Einstein metric to high precision in a bounded region containing the singular orbit or "bolt", which is then perturbed to a genuine Einstein metric using fixed-point methods. At the boundary of this region, the latter metric is sufficiently close to hyperbolic space for us to show that it indeed extends to a complete, asymptotically hyperbolic Einstein metric.
Paper Structure (32 sections, 9 theorems, 165 equations, 2 figures, 3 tables)

This paper contains 32 sections, 9 theorems, 165 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction and preliminaries
  2. Introduction
  3. Solution starting from the singular orbit
  4. Extending the metric to infinity
  5. Setup and notation
  6. Known examples and results for subsystems
  7. Summary of numerical results for negative Einstein metrics
  8. Plan of the paper
  9. Negative Einstein metrics in the conformally compact formalism
  10. Proof of existence : asymptotic hyperbolicity near infinity
  11. Estimates under assumptions C and D
  12. Singular initial value problem and approximate solution
  13. Proof of existence : fixed-point argument starting from the singular orbit
  14. $C^1$-type estimate
  15. Linear inhomogeneous initial value problem and $C^0$ estimate
  16. ...and 17 more sections

Key Result

Theorem 3.1

For $t\in[a,b]$, if $\beta\geq 0$ and $u$ satisfies then

Figures (2)

  • Figure 1: Numerical evidence for the existence of complete negative Einstein metrics in the parameter plane $(h,b_1)$ for an $\dot a(0)=2$ bolt, corresponding to the total space $M=\mathcal{O}(-4)$. The dark shaded region corresponds to likely complete Einstein metrics. The region with negative $b_1$ is not shown, as there is a symmetry $b_1\leftrightarrow -b_1$, $b\leftrightarrow c$.
  • Figure 2: Plot of the warping functions of the Einstein metric constructed in § \ref{['computersection']}, with $h=1.5$, $b_1=0.1$. The cutoff time where the fixed-point construction of §\ref{['fixedpointsection']} meets the asymptotic hyperbolicity estimates of §\ref{['infinitysection']} is indicated by the brown dashed line. The numerics are produced using an order 8 Runge--Kutta solver.

Theorems & Definitions (11)

  • Theorem 3.1
  • Lemma 3.2
  • proof
  • Lemma 5.1
  • proof
  • Lemma 5.2
  • Lemma 5.3
  • Theorem 5.4
  • Lemma 5.5
  • Theorem 6.1
  • ...and 1 more