Gravitational instantons, self-duality and geometric flows

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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 × M₃ 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.