Asymptotically hyperbolic extensions and an analogue of the Bartnik mass

Pacheco, Armando J. Cabrera; Cederbaum, Carla; McCormick, Stephen

doi:10.1016/j.geomphys.2018.06.010

Cited by 13 publications

(15 citation statements)

References 20 publications

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“…This question is related to the proof of (ii) in [14] and similarly to the proof of Proposition 1.1. This is also related to the estimate of the generalized Bartnik mass by Cabrera Pacheco, Cederbaum and McCormick [4]. In §2 we will sketch a proof of a modified version of Trudinger's theorem [26].…”

Section: Introductionmentioning

confidence: 88%

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Singular metrics with negative scalar curvature

Man-Chuen¹,

Lee²,

Tam³

2021

Preprint

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Motivated by the work of Li and Mantoulidis [14], we study singular metrics which are uniformly Euclidean (L ∞ ) on a compact manifold M n (n ≥ 3) with negative Yamabe invariant σ(M ). It is well-known that if g is a smooth metric on M with unit volume and with scalar curvature S(g) ≥ σ(M ), then g is Einstein. We show, in all dimensions, the same is true for metrics with edge singularities with cone angles ≤ 2π along codimension-2 submanifolds. We also show in three dimension, if the Yamabe invariant of connected sum of two copies of M attains its minimum, then the same is true for L ∞ metrics with isolated point singularities.

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Section: Introductionmentioning

confidence: 88%

“…Using the result in §5 below and the method in [19], see also [4], one can construct a metric on the upper hemisphere with scalar curvature bounded below by σ so that the boundary is isometric to g ε | S in the previous lemma and is minimal. However, one cannot obtain a good estimate for the volume.…”

Section: Point Singularitymentioning

confidence: 99%

Singular metrics with negative scalar curvature

Man-Chuen¹,

Lee²,

Tam³

2021

Preprint

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“…In the Riemannian setting, Mantoulidis and Schoen [53] proved that the Bartnik mass M B (or M outer ) of a horizon is given by the irreducible mass (9). Their result is generalized to hyperbolic case (10) by Pacheco et al [51]. M outer is technically easier to work with than M B .…”

Section: The Bartnik-bray Quasilocal Massmentioning

confidence: 99%

“…Finally, we shall point out that as compared to M outer , the AdS version is much less studied in the literature. It was not studied only until recently been first proposed in [51].…”

Section: The Bartnik-bray Quasilocal Massmentioning

confidence: 99%

Outer entropy equals Bartnik-Bray inner mass and the gravitational ant conjecture

Wang

2020

Phys. Rev. D

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Entropy and energy are found to be closely tied on our quest for quantum gravity. We point out an interesting connection between the recently proposed outer entropy, a coarse-grained entropy defined for a compact spacetime domain motivated by the holographic duality, and the Bartnik-Bray quasilocal mass long known in the mathematics community. In both scenarios, one seeks an optimal spacetime fill-in of a given closed, connected, spacelike, codimension-two boundary. We show that for an outer-minimizing mean-convex surface, the Bartnik-Bray inner mass matches exactly with the irreducible mass corresponding to the outer entropy. The equivalence implies that the area laws derived from the outer entropy are mathematically equivalent as the monotonicity property of the quasilocal mass. It also gives rise to new bounds between entropy and the gravitational energy, which naturally gives the gravitational counterpart to Wall's ant conjecture. We also observe that the equality can be achieved in a conformal flow of metrics, which is structurally similar to the Ceyhan-Faulkner proof of the ant conjecture. We compute the small sphere limit of the outer entropy and it is proportional to the bulk stress tensor as one would expect for a quasilocal mass. Last, we discuss some implications of taking quantum matter into consideration in the semiclassical setting.

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“…The collar C used here is very similar to what is used in [18], which was inspired by the work of Mantoulidis and Schoen [17]. In fact, these kinds of collars have proven quite useful in relation to the Bartnik mass [9,8], and quasi-local mass quantities more generally [20]. Figure 1.…”

Section: Construction Of Collarsmentioning

confidence: 99%

Gluing Bartnik extensions, continuity of the Bartnik mass, and the equivalence of definitions

McCormick

2020

Pacific J. Math.

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In the context of the Bartnik mass, there are two fundamentally different notions of an extension of some compact Riemannian manifold (Ω, γ) with boundary. In one case, the extension is taken to be a manifold without boundary in which (Ω, γ) embeds isometrically, and in the other case the extension is taken to be a manifold with boundary where the boundary data is determined by ∂Ω.We give a type of convexity condition under which we can say both of these types of extensions indeed yield the same value for the Bartnik mass. Under the same hypotheses we prove that the Bartnik mass varies continuously with respect to the boundary data. This also provides a method to use estimates for the Bartnik mass of constant mean curvature (CMC)

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Asymptotically hyperbolic extensions and an analogue of the Bartnik mass

Cited by 13 publications

References 20 publications

Singular metrics with negative scalar curvature

Singular metrics with negative scalar curvature

Outer entropy equals Bartnik-Bray inner mass and the gravitational ant conjecture

Gluing Bartnik extensions, continuity of the Bartnik mass, and the equivalence of definitions

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