Gravitational instantons, self-duality and geometric flows

Abstract

We discuss four-dimensional "spatially homogeneous" gravitational instantons. These are self-dual solutions of Euclidean vacuum Einstein's equations with potentially non-vanishing cosmological constant. They are endowed with a product structure R \times M_3 leading to a natural foliation into three-dimensional subspaces evolving in Euclidean time. For a large class of three-dimensional subspaces, the dynamics coincides with the geometric flow on the three-dimensional homogeneous slice, driven by the Ricci tensor plus an so(3) gauge connection. The metric on the three-dimensional space is related to the vielbein of the three-dimensional subspace, while the gauge field is inherited from the anti-self-dual component of the four-dimensional Levi--Civita connection.

14 pages

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Cited by in corpus (8)

• G3-homogeneous gravitational instantons
• Exact helicoidal and catenoidal solutions in Einstein-Maxwell theory
• Gravitational duality, topologically massive gravity and holographic fluids
• Self-dual gravitational instantons and geometric flows of all Bianchi types
• NUT Solutions in Einstein-Maxwell-scalar-Gauss-Bonnet Gravity
• Geometric Structure of Higher-Dimensional Spheres
• New Lorentzian Taub-NUT and Euclidean Eguchi-Hanson Solutions in $f(R)$ gravity
• Bianchi type A hyper-symplectic and hyper-Kähler metrics in 4D