Riemannian 4-manifolds obeying Einstein's equations play a significant role in attempts to understand quantum gravity, from black hole thermodynamics to string theory. I will describe how the Ricci flow provides a natural gradient flow for the Euclidean EinsteinHilbert action which in the context of string theory may be thought of as an R-G flow. I will show how this flow may be applied to the Euclidean Schwarzschild and Taub-NUT metrics and discuss the mathematical and physical problems this poses.