Renormalized Volume

Abstract

We develop a universal distributional calculus for regulated volumes of metrics that are singular along hypersurfaces. When the hypersurface is a conformal infinity we give simple integrated distribution expressions for the divergences and anomaly of the regulated volume functional valid for any choice of regulator. For closed hypersurfaces or conformally compact geometries, methods from a previously developed boundary calculus for conformally compact manifolds can be applied to give explicit holographic formulae for the divergences and anomaly expressed as hypersurface integrals over local quantities (the method also extends to non-closed hypersurfaces). The resulting anomaly does not depend on any particular choice of regulator, while the regulator dependence of the divergences is precisely captured by these formulae. Conformal hypersurface invariants can be studied by demanding that the singular metric obey, smoothly and formally to a suitable order, a Yamabe type problem with boundary data along the conformal infinity. We prove that the volume anomaly for these singular Yamabe solutions is a conformally invariant integral of a local Q-curvature that generalizes the Branson Q-curvature by including data of the embedding. In each dimension this canonically defines a higher dimensional generalization of the Willmore energy/rigid string action. Recently Graham proved that the first variation of the volume anomaly recovers the density obstructing smooth solutions to this singular Yamabe problem; we give a new proof of this result employing our boundary calculus. Physical applications of our results include studies of quantum corrections to entanglement entropies.Comment: 31 pages, LaTeX, 5 figures, anomaly formula generalized to any bulk geometry, improved discussion of hypersurfaces with boundar

Volume renormalization for singular Yamabe metrics

“…The main purpose of this paper is to show that such an energy can be constructed by renormalizing the volume of the singular Yamabe metric. After we described this work to Gover and Waldron and showed them our proof, they posted [GoW4], which also discusses volume renormalization for singular Yamabe metrics and proves the same result.…”

“…Changing from r to ρ is equivalent to changing the choice of background metric from g to Ω 2 g with Ω = |dρ| g , since |dρ| Ω 2 g = 1 so that ρ is the distance to Σ in the metric Ω 2 g. So Proposition 2.1 implies that the coefficient of log 1 ǫ (the energy) is independent of the choice of ρ. If one takes ρ = u, then the coefficients of all the divergent terms are integrals of local invariants of g, just like for ρ = r, since the Taylor expansion of u is locally determined by g. In [GoW4], closed formulae are derived for all the coefficients c 0 , . .…”

“…Our energy is the coefficient of the log 1 ǫ term in the renormalized volume expansion. It is the integral of v (n) and is invariant under conformal rescalings of g. In [GoW4], a version of Q-curvature is defined for the singular Yamabe setting and the energy is shown to equal the integral of the Q-curvature. The fact that the variation of the energy with respect to Σ is the log term coefficient in the solution u is an analog of the result of [HSS], [GH] that in the Poincaré-Einstein case with even-dimensional boundary, the metric variation of the log term coefficient in the volume expansion is a constant multiple of the ambient obstruction tensor, the log term coefficient in the expansion of the Poincaré-Einstein metric itself.…”

A calculus for conformal hypersurfaces and new higher Willmore energy functionals

“…[28,34]. Recently energy functionals for these objects have been constructed from conformal anomalies in a renormalised volume expansion [20] (see also [19]). A second main aim here is to apply tools developed in [17,18] to provide a construction of manifestly conformally invariant energies with the same leading order functional gradient (with respect to variation of embedding) as the anomaly functionals.…”

“…A second main aim here is to apply tools developed in [17,18] to provide a construction of manifestly conformally invariant energies with the same leading order functional gradient (with respect to variation of embedding) as the anomaly functionals. Not only do these new energies yield alternative conformally invariant higher Willmore equation, the nature of these suggests they will also be useful for analysing and even altering the functionals in [19,20]. Alterations may be useful because the positivity of these higher "energies" is not established.…”