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

2025/04/30 by Wang, Qiu Shi · 1 citation
#53C25 #65D15 #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2504.21644

Abstract

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 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.

Citations

• Cohomogeneity one 4-dimensional gradient Ricci solitons
• Initial value problems on cohomogeneity one manifolds, I
• Einstein metrics on the Ten-Sphere
• Arb: Efficient Arbitrary-Precision Midpoint-Radius Interval Arithmetic
• Bianchi IX Self-dual Einstein Metrics and Singular G₂ Manifolds

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• Poincaré-Einstein 4-manifolds with conformally Kähler geometry

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