Self-dual gravitational instantons and geometric flows of all Bianchi types

We investigate four-dimensional, self-dual gravitational instantons endowed with a product structure R × M3, where M3 is a homogeneous threedimensional manifold of Bianchi type.We analyze the general conditions under which Euclidean-time evolution in the gravitational instanton can be identified with a geometric flow of a metric onM3. This includes both unimodular and non-unimodular groups, and the corresponding geometric flow is a general Ricci plus Yang-Mills flow accompanied by a diffeomorphism.