  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.
