Renormalized volume and the evolution of APEs

Abstract

Abstract We study the evolution of the renormalized volume functional for even-dimensional asymptotically Poincaré-Einstein metrics (M, g) under normalized Ricci flow. In particular, we prove that where S(g(t)) is the scalar curvature for the evolving metric g(t). This implies that if S +n(n − 1) ≥ 0 at t = 0, then RenV(Mn , g(t)) decreases monotonically. For odd-dimensional asymptotically Poincaré-Einstein metrics with vanishing obstruction tensor,we find that the conformal anomaly for these metrics is constant along the flow. We apply our results to the Hawking-Page phase transition in black hole thermodynamics.We compute renormalized volumes for the Einstein 4-metrics sharing the conformal infinity of an AdS-Schwarzschild black hole. We compare these to the free energies relative to thermal hyperbolic space, as originally computed by Hawking and Page using a different regularization technique, and find that they are equal.