Volume Comparison of Conformally Compact Manifolds with Scalar Curvature R ≥ −n (n − 1)

Abstract

In this paper, we use the normalized Ricci-DeTurk flow to prove a stability result for strictly stable conformally compact Einstein manifolds. As an application, we show a local volume comparison of conformally compact manifolds with scalar curvature R ≥ −n (n − 1) and also the rigidity result when certain relative volume is zero.Resumé. Dans cet article, nous utilisons le flot de Ricci-DeTurk normalisé pour prouver la stabilité des variétés d'Einstein strictement stables et conformément compactes. En tant qu'application, nous montrons une comparaison de volume local pour les variétés conformément compactes dont la courbure scalaire satisfait R ≥ −n(n − 1). Nous donnonségalement un résultat de rigidité lorsque certain volume relative est nul.

General bounds on holographic complexity

“…We established this result rigorously in broad generality in four bulk dimensions, assuming the weak energy condition. We have also established a weaker statement in other dimensions, that the vacuum-subtracted volume is positive in spacetimes with sufficient symmetry or in perturbations of the AdS vacuum [44] (again assuming the weak energy condition). The more general statement for arbitrary spacetimes satisfying the weak energy condition would follow from a modification of a well-known mathematical conjecture [43,66] from compact to conformally compact manifolds.…”

“…The generalization to arbitrary dimensions can be proven from a generalization of a well-known mathematical conjecture by Schoen [43]; without relying on conjectures, we give a proof under the assumption of spherical or planar symmetry that C F is positive (again assuming the WCC). We also highlight an existing mathematical theorem [44] proving that sufficiently small (but finite) WCC-preserving deformations of pure AdS result in positive C F , thus establishing in all dimensions that pure AdS is a local minimum of C F within the space of JHEP01(2022)040 WCC-preserving asymptotically AdS spacetimes. We consider the combination of our results, Schoen's conjecture and the small-deformation results [44] to be strong evidence in favor of the higher-dimensional generalization of the positive complexity theorem.…”

“…We also highlight an existing mathematical theorem [44] proving that sufficiently small (but finite) WCC-preserving deformations of pure AdS result in positive C F , thus establishing in all dimensions that pure AdS is a local minimum of C F within the space of JHEP01(2022)040 WCC-preserving asymptotically AdS spacetimes. We consider the combination of our results, Schoen's conjecture and the small-deformation results [44] to be strong evidence in favor of the higher-dimensional generalization of the positive complexity theorem.…”

“…In this result δg need not be infinitesimal, as long as it is sufficiently small in an appropriate norm [44].…”

“…Theorem 3 ([44]). Let δg be a metric perturbation to global pure AdS d+1 , where the perturbation approaches zero sufficiently quickly to not affect the asymptotics [44]. A sufficiently small such δg satisfying the WCC cannot increase the volume of static-anchored maximalvolume slices.…”