Geometric flows and black hole entropy

Perelman has given a gradient formulation for the Ricci flow, introducing an ``entropy function'' which increases monotonically along the flow.We pursue a thermodynamic analogy and apply Ricci flow ideas to general relativity. We investigate whether Perelman's entropy is related to (Bekenstein-Hawking)geometric entropy as familiar from black hole thermodynamics. From a study of the fixed points of the flow we conclude that Perelman entropy is not connected to geometric entropy. However, we notice that there is a very similar flow which DOES appear to be connected to geometric entropy. The new flow may find applications in black hole physics suggesting for instance, new approaches to the Penrose inequality.