Gravitational instantons, self-duality, and geometric flows

We discuss four-dimensional "spatially homogeneous" gravitational instantons. These are self-dual solutions of Euclidean vacuum Einstein equations. They are endowed with a product structure R×M3 leading to a foliation into three-dimensional subspaces evolving in Euclidean time. For a large class of homogeneous subspaces, the dynamics coincides with a geometric flow on the three-dimensional slice, driven by the Ricci tensor plus an so(3) gauge connection. The flowing metric is related to the vielbein of the subspace, while the gauge field is inherited from the anti-self-dual component of the four-dimensional Levi-Civita connection.