abstract: Instanton solutions of Euclidean non-relativistic gravity in four dimensions are described by geometric flows in three dimensions driven by certain combinations of the Cotton and Ricci tensors. The deformation curvature terms have competing behavior leading to a variety of fixed points. The instantons are finite-action solutions interpolating between any two such fixed points. Each of these fixed points is itself a vacuum solution of topological massive gravity.
This web of relations will be presented and applied to the special case of Bianchi IX homogeneous geometries, where the Ricci-Cotton flow reduces to an autonomous system of ordinary differential equations. The properties of that system are studied in detail allowing for a full classification of the corresponding instantons.