In this paper we present some results on a family of geometric flows introduced by J. P. Bourguignon in [4] that generalize the Ricci flow. For suitable values of the scalar parameter involved in these flows, we prove short time existence and provide curvature estimates. We also state some results on the associated solitons.

Given a compact three-manifold together with a Riemannian metric, we prove the short-time existence of a solution to the renormalization group flow, truncated at the second order term, under a suitable hypothesis on the sectional curvature of the initial metric.