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.
@article{bwmeta1.element.doi-10_1515_geofl-2015-0007,
author = {Eric Bahuaud and Rafe Mazzeo and Eric Woolgar},
title = {Renormalized volume and the evolution of APEs},
journal = {Geometric Flows},
volume = {1},
year = {2015},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_1515_geofl-2015-0007}
}
Eric Bahuaud; Rafe Mazzeo; Eric Woolgar. Renormalized volume and the evolution of APEs. Geometric Flows, Tome 1 (2015) . http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_1515_geofl-2015-0007/
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