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 $\mathbb{R}\times {\mathcal{M}}_3$ 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 $\mathfrak{s}\mathfrak{o}(3)$ 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.