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

Qiu Shi Wang

arxiv: 2504.21644 · v2 · submitted 2025-04-30 · 🧮 math.DG

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

Qiu Shi Wang This is my paper

Pith reviewed 2026-05-22 18:07 UTC · model grok-4.3

classification 🧮 math.DG

keywords negative Einstein metricsSU(2)-invariant metricsconformally compact manifoldsasymptotically hyperbolic metricscomplex line bundlesrigorous numericsfixed-point methodstriaxial symmetry

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The pith

A two-parameter family of new SU(2)-invariant negative Einstein metrics exists on the complex line bundle O(-4) over CP1.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper aims to establish the existence of a two-parameter family of triaxial SU(2)-invariant complete negative Einstein metrics on the complex line bundle O(-4) over CP1. These metrics are conformally compact and asymptotically hyperbolic but neither Kähler nor self-dual. The construction begins with high-precision numerical approximation of the metric near the bolt and then uses fixed-point methods to produce an exact solution that extends to a complete asymptotically hyperbolic metric. If this holds, it supplies new explicit examples of negative Einstein metrics with these symmetries and shows how rigorous numerics can resolve existence questions that resist purely analytic treatment.

Core claim

We construct a 2-parameter family of new triaxial SU(2)-invariant complete negative Einstein metrics on the complex line bundle O(-4) over CP1. The metrics are conformally compact and 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 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.

What carries the argument

Rigorous numerical approximation of an Einstein metric near the bolt, followed by fixed-point perturbation to obtain an exact solution whose boundary data is close enough to hyperbolic space for a complete asymptotically hyperbolic extension.

If this is right

The construction yields new examples of conformally compact negative Einstein manifolds that carry SU(2) symmetry and are neither Kähler nor self-dual.
The resulting metrics are complete and asymptotically hyperbolic.
The method succeeds for triaxial (as opposed to biaxial) symmetry classes.
Computer-assisted approximation plus fixed-point perturbation can establish existence for negative Einstein metrics on complex line bundles.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same numerical-plus-fixed-point strategy could be tested on line bundles of other degrees or on manifolds with different symmetry groups.
The new metrics supply concrete points in the moduli space of asymptotically hyperbolic Einstein structures that can be studied for stability or deformation properties.
Refinements in the numerical precision or in the choice of the bounded region might allow construction of metrics with additional continuous parameters.

Load-bearing premise

The numerically computed approximate metric is sufficiently close in a suitable norm to a genuine solution inside the bounded region so that the fixed-point argument produces an exact Einstein metric whose boundary data is close enough to hyperbolic space to guarantee a complete asymptotically hyperbolic extension.

What would settle it

A more precise error analysis or independent computation showing that the numerical approximation lies outside the ball in which the fixed-point theorem guarantees a solution, or that the perturbed metric fails to approach hyperbolic space sufficiently closely at the boundary of the region.

Figures

Figures reproduced from arXiv: 2504.21644 by Qiu Shi Wang.

**Figure 1.** Figure 1: Numerical evidence for the existence of complete negative Einstein metrics in the parameter plane (h, b1) for an ˙a(0) = 2 bolt, corresponding to the total space M = O(−4). The dark shaded region corresponds to likely complete Einstein metrics. The region with negative b1 is not shown, as there is a symmetry b1 ↔ −b1, b ↔ c. • When the singular orbit is S 2 and the principal orbit is SU(2) (bolt, M = O(−… view at source ↗

**Figure 2.** Figure 2: Plot of the warping functions of the Einstein metric constructed in §6, with h = 1.5, b1 = 0.1. The cutoff time where the fixed-point construction of §5 meets the asymptotic hyperbolicity estimates of §3 is indicated by the brown dashed line. The numerics are produced using an order 8 Runge– Kutta solver [PITH_FULL_IMAGE:figures/full_fig_p031_2.png] view at source ↗

read the original 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 neither K\"ahler 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper gives a computer-assisted construction of a 2-parameter family of triaxial SU(2)-invariant negative Einstein metrics on O(-4) that are new and neither Kähler nor self-dual.

read the letter

The main takeaway is a construction of a 2-parameter family of triaxial SU(2)-invariant complete negative Einstein metrics on the bundle O(-4) over CP1. These metrics are conformally compact and avoid the Kähler and self-dual cases that dominate earlier examples in this symmetry class. The proof splits into a high-precision numerical approximation of an Einstein metric near the bolt, a fixed-point correction to make it exact, and an extension to an asymptotically hyperbolic end that stays close to hyperbolic space at infinity. This combination produces concrete new examples where closed-form solutions have been hard to find. The numerical-plus-perturbation strategy is a recognized tool for nonlinear geometric PDEs when analytic methods stall, and the paper applies it to a symmetry setting that has seen limited triaxial results before. If the error controls hold, it adds usable examples to the catalog of invariant Einstein metrics. The potential weak point is the validation of the numerical error in the exact Banach norm needed for the contraction mapping. The abstract states that the approximation is close enough for the fixed-point theorem to apply, but the argument requires explicit bounds on the inverse of the linearized Einstein operator in the chosen weighted spaces. Without those interval-arithmetic checks or operator-norm estimates shown in detail, the threshold for contraction remains unconfirmed, and the same gap affects the boundary matching. This is a standard concern for computer-assisted proofs rather than a fatal flaw, but it is the part that needs the most referee attention. The paper targets specialists in geometric analysis who work on Einstein metrics with group actions or who use rigorous numerics for existence questions. A reader already familiar with SU(2)-invariant constructions or fixed-point methods in manifolds will get the most out of the details. It has enough specificity and technical substance to merit a full referee process instead of a desk rejection. I would send it to peer review, with the main request being a clear presentation of the error bounds in the precise function spaces used for the perturbation step.

Referee Report

1 major / 2 minor

Summary. The paper constructs a 2-parameter family of triaxial SU(2)-invariant complete negative Einstein metrics on the complex line bundle O(-4) over CP^1. These metrics are conformally compact and neither Kähler nor self-dual. The proof proceeds by using rigorous high-precision numerics to obtain an approximate Einstein metric on a compact region containing the bolt, applying a fixed-point theorem to perturb it to an exact solution, and then extending the metric to a complete asymptotically hyperbolic Einstein metric by matching to hyperbolic space at the boundary of the region.

Significance. If the error bounds and contraction estimates are rigorously validated, the result supplies new explicit examples of negative Einstein metrics with SU(2) symmetry on a non-compact 4-manifold, illustrating the viability of computer-assisted methods that combine numerical approximation with analytic perturbation techniques. Such constructions can serve as concrete test cases for questions about the moduli space of Einstein metrics and the behavior of asymptotically hyperbolic solutions.

major comments (1)

[Section describing the rigorous numerics and fixed-point argument] The fixed-point perturbation step requires that the residual ||Ein(g_approx)||_X be strictly less than 1/(2C) where C is the operator norm of the inverse of the linearized Einstein operator D Ein(g_approx) from Y to X. The manuscript states that the numerical approximation is “sufficiently close” but does not supply explicit interval-arithmetic bounds on this residual and on C in the precise weighted Hölder (or Sobolev) spaces adapted to the SU(2) symmetry and bolt regularity. This verification is load-bearing for the existence claim.

minor comments (2)

[Abstract] The abstract and introduction could include a brief statement of the achieved numerical precision (e.g., number of digits or interval width) and the software library employed, to aid reproducibility.
[Fixed-point section] Notation for the Banach spaces X and Y used in the contraction mapping should be introduced once and used consistently when stating the smallness condition on the residual.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying the need for more explicit verification in the fixed-point argument. We address the major comment below and will incorporate the requested details in a revised version.

read point-by-point responses

Referee: [Section describing the rigorous numerics and fixed-point argument] The fixed-point perturbation step requires that the residual ||Ein(g_approx)||_X be strictly less than 1/(2C) where C is the operator norm of the inverse of the linearized Einstein operator D Ein(g_approx) from Y to X. The manuscript states that the numerical approximation is “sufficiently close” but does not supply explicit interval-arithmetic bounds on this residual and on C in the precise weighted Hölder (or Sobolev) spaces adapted to the SU(2) symmetry and bolt regularity. This verification is load-bearing for the existence claim.

Authors: We agree that the current presentation does not include sufficiently explicit interval-arithmetic bounds on the residual and on the operator norm C in the weighted Hölder spaces respecting the SU(2) symmetry and bolt conditions. In the revision we will add a new subsection (or appendix) that reports the concrete interval enclosures obtained for ||Ein(g_approx)||_X together with a rigorous upper bound for the norm of the inverse of the linearized operator, computed via validated numerics on a finite-dimensional projection plus tail estimates. These bounds will be shown to satisfy the strict inequality required for the contraction mapping, thereby closing the existence argument. revision: yes

Circularity Check

0 steps flagged

No circularity: existence via rigorous numerics plus fixed-point theorem

full rationale

The derivation proceeds by first computing a high-precision approximate Einstein metric on a compact region containing the bolt using rigorous numerics, then invoking a fixed-point theorem in suitable Banach spaces to obtain an exact solution whose boundary values are close enough to hyperbolic space to guarantee a complete asymptotically hyperbolic extension. This chain relies on external analytic tools (contraction mapping, weighted Hölder/Sobolev norms) and independently validated numerical error control rather than any self-definitional reduction, fitted parameter renamed as prediction, or load-bearing self-citation. No equation or step in the provided abstract or description reduces the claimed existence result to its own inputs by construction.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The construction rests on standard differential-geometric background (Einstein equation, SU(2) invariance, conformal compactness) plus the technical assumption that the numerical approximation lies inside the contraction ball of the fixed-point map. No new physical entities are introduced.

free parameters (1)

two continuous parameters of the family
The family is parameterized by two real numbers; these are free parameters that label distinct metrics in the constructed family.

axioms (2)

domain assumption The Einstein equation Ric(g) = -3g admits solutions with the given symmetry and asymptotic behavior.
Standard background assumption in the field of Einstein metrics; invoked implicitly when stating the target equation.
standard math Fixed-point theorems apply once the numerical error is below a computable threshold.
The perturbation step relies on a contraction-mapping or implicit-function theorem in appropriate Banach spaces.

pith-pipeline@v0.9.0 · 5629 in / 1551 out tokens · 37383 ms · 2026-05-22T18:07:12.928051+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We construct a 2-parameter family of new triaxial SU(2)-invariant complete negative Einstein metrics on the complex line bundle O(-4) over CP1 ... using rigorous numerics to produce an approximate Einstein metric ... then perturbed to a genuine Einstein metric using fixed-point methods.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

the tangential Einstein equation ... conservation law ... singular initial value problem ... fixed-point theorem 5.4

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

23 extracted references · 23 canonical work pages

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[1] [1]

Hitchin, The geometry and dynamics of magnetic monopoles , M

Michael Atiyah and Nigel J. Hitchin, The geometry and dynamics of magnetic monopoles , M. B. Porter Lectures, Princeton University Press, Princeton, NJ, 1988

work page 1988

[2] [2]

´Elie Cartan, vol

Lionel B´ erard Bergery,Sur de nouvelles vari´ et´ es riemanniennes d’Einstein, Institut ´Elie Cartan, 6, Inst. ´Elie Cartan, vol. 6, Univ. Nancy, Nancy, 1982, pp. 1–60

work page 1982

[3] [3]

Christoph B¨ ohm, Inhomogeneous Einstein metrics on low-dimensional spheres and other low- dimensional spaces, Invent. Math. 134 (1998), no. 1, 145–176

work page 1998

[4] [4]

Timothy Buttsworth and Liam Hodgkinson, Computationally-assisted proof of a novel O(3) × O(10)- invariant Einstein metric on S12, https://arxiv.org/abs/2412.01184, 2024

work page arXiv 2024

[5] [5]

Chang and Yuxin Ge, On conformally compact Einstein manifolds , Rev

Sun-Yung A. Chang and Yuxin Ge, On conformally compact Einstein manifolds , Rev. Un. Mat. Argentina 64 (2022), no. 1, 199–213

work page 2022

[6] [6]

Cvetiˇ c, G

M. Cvetiˇ c, G. W. Gibbons, H. L¨ u, and C. N. Pope,Bianchi IX self-dual Einstein metrics and singular G2 manifolds, Classical Quantum Gravity 20 (2003), no. 19, 4239–4268

work page 2003

[7] [7]

Dancer and Ian A

Andrew S. Dancer and Ian A. B. Strachan, K¨ ahler-Einstein metrics withSU(2) action, Math. Proc. Cambridge Philos. Soc. 115 (1994), no. 3, 513–525

work page 1994

[8] [8]

Dancer and McKenzie Y

Andrew S. Dancer and McKenzie Y. Wang, The cohomogeneity one Einstein equations from the Hamiltonian viewpoint, J. Reine Angew. Math. 524 (2000), 97–128

work page 2000

[9] [9]

Patrick Donovan, Cohomogeneity one 4-dimensional gradient Ricci solitons , https://arxiv.org/ abs/2503.15033, 2025

work page arXiv 2025

[10] [10]

Hanson, Self-dual solutions to Euclidean gravity , Ann

Tohru Eguchi and Andrew J. Hanson, Self-dual solutions to Euclidean gravity , Ann. Physics 120 (1979), no. 1, 82–106

work page 1979

[11] [11]

Eschenburg and McKenzie Y

J.-H. Eschenburg and McKenzie Y. Wang, The initial value problem for cohomogeneity one Einstein metrics, J. Geom. Anal. 10 (2000), no. 1, 109–137

work page 2000

[12] [12]

Lorenzo Foscolo and Mark Haskins, New G2-holonomy cones and exotic nearly K¨ ahler structures on S6 and S3 × S3, Ann. of Math. (2) 185 (2017), no. 1, 59–130

work page 2017

[13] [13]

G. W. Gibbons and C. N. Pope, The positive action conjecture and asymptotically Euclidean metrics in quantum gravity , Comm. Math. Phys. 66 (1979), no. 3, 267–290

work page 1979

[14] [14]

Robin Graham and John M

C. Robin Graham and John M. Lee, Einstein metrics with prescribed conformal infinity on the ball , Adv. Math. 87 (1991), no. 2, 186–225

work page 1991

[15] [15]

Hitchin, Twistor spaces, Einstein metrics and isomonodromic deformations , J

Nigel J. Hitchin, Twistor spaces, Einstein metrics and isomonodromic deformations , J. Differential Geom. 42 (1995), no. 1, 30–112

work page 1995

[16] [16]

Johansson, Arb: efficient arbitrary-precision midpoint-radius interval arithmetic , IEEE Transac- tions on Computers 66 (2017), 1281–1292

F. Johansson, Arb: efficient arbitrary-precision midpoint-radius interval arithmetic , IEEE Transac- tions on Computers 66 (2017), 1281–1292

work page 2017

[17] [17]

Newman, L

E. Newman, L. Tamburino, and T. Unti, Empty-space generalization of the Schwarzschild metric , J. Mathematical Phys. 4 (1963), 915–923

work page 1963

[18] [18]

Jan Nienhaus and Matthias Wink, Einstein metrics on the Ten-Sphere , https://arxiv.org/abs/ 2303.04832, 2023

work page arXiv 2023

[19] [19]

Page, A compact rotating gravitational instanton, Physics Letters B 79 (1978), no

Don N. Page, A compact rotating gravitational instanton, Physics Letters B 79 (1978), no. 3, 235–238

work page 1978

[20] [20]

2, 249–251

, Taub-NUT instanton with an horizon , Physics Letters B 78 (1978), no. 2, 249–251

work page 1978

[21] [21]

Pedersen, Eguchi-Hanson metrics with cosmological constant, Classical Quantum Gravity 2 (1985), no

H. Pedersen, Eguchi-Hanson metrics with cosmological constant, Classical Quantum Gravity 2 (1985), no. 4, 579–587

work page 1985

[22] [22]

Luigi Verdiani and Wolfgang Ziller, Initial value problems on cohomogeneity one manifolds, I , https: //arxiv.org/abs/2412.06058, 2024

work page arXiv 2024

[23] [23]

Wang and Wolfgang Ziller, Existence and nonexistence of homogeneous Einstein metrics, Invent

McKenzie Y. Wang and Wolfgang Ziller, Existence and nonexistence of homogeneous Einstein metrics, Invent. Math. 84 (1986), no. 1, 177–194. Mathematical Institute, University of Oxford, Oxford, OX2 6GG, United Kingdom Email address : wangqs@maths.ox.ac.uk

work page 1986