Higher dimensional black hole initial data with prescribed boundary metric

Pacheco, Armando J. Cabrera; Miao, Pengzi

doi:10.4310/mrl.2018.v25.n3.a10

Cited by 18 publications

(12 citation statements)

References 12 publications

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“…Similar to the previous lemma, high regularity decay estimates follows by Lemma 3.7 and the definition ofg ij . Combining results in this section and arguments in [12,Section 3], one knows that g can be connected to a round metric within the space of positive scalar curvature metrics on S n . Therefore, repeating the proof in [12], we know that the conclusion of [12, Theorem 1.2] holds for such a metric g.…”

Section: Asymptotic Behaviorsmentioning

confidence: 76%

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Minimal hypersurfaces and boundary behavior of compact manifolds with nonnegative scalar curvature

Miao

2019

J. Differential Geom.

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On a compact Riemannian manifold with boundary having positive mean curvature, a fundamental result of Shi and Tam states that, if the manifold has nonnegative scalar curvature and if the boundary is isometric to a strictly convex hypersurface in the Euclidean space, then the total mean curvature of the boundary is no greater than the total mean curvature of the corresponding Euclidean hypersurface. In 3-dimension, Shi-Tam's result is known to be equivalent to the Riemannian positive mass theorem.In this paper, we provide a supplement to Shi-Tam's result by including the effect of minimal hypersurfaces on the boundary. More precisely, given a compact manifold Ω with nonnegative scalar curvature, assuming its boundary consists of two parts, Σ H and Σ O , where Σ H is the union of all closed minimal hypersurfaces in Ω and Σ O is isometric to a suitable 2-convex hypersurface Σ in a spatial Schwarzschild manifold of positive mass m, we establish an inequality relating m, the area of Σ H , and two weighted total mean curvatures of Σ O and Σ. In 3-dimension, the inequality has implications to both isometric embedding and quasi-local mass problems. In a relativistic context, our result can be interpreted as a quasi-local mass type quantity of Σ O being greater than or equal to the Hawking mass of Σ H . We further analyze the limit of such quasi-local mass quantity associated with suitably chosen isometric embeddings of large coordinate spheres of an asymptotically flat 3-manifold M into a spatial Schwarzschild manifold. We show that the limit equals the ADM mass of M . It follows that our result on the compact manifold Ω is equivalent to the Riemannian Penrose inequality.

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Section: Asymptotic Behaviorsmentioning

confidence: 76%

“…12) where h is the second fundamental form of Σ r . Taking trace of (6.12) gives Now we apply [23, Lemma 2.5] to get another Minkowski type identity.…”

mentioning

confidence: 99%

Minimal hypersurfaces and boundary behavior of compact manifolds with nonnegative scalar curvature

Miao

2019

J. Differential Geom.

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“…This construction has proven to be useful to obtain Bartnik mass estimates. Relevant and related results include those in [6,17,5,4,16]; for a survey on this topic see [3].…”

Section: Introduction and Resultsmentioning

confidence: 99%

“…One can explicitly define this mollification as in [6,Equation (4.19)]. In addition, one can check that…”

Section: Gluing Methods For Rotationally Symmetric Electrically Charg...mentioning

confidence: 99%

Asymptotically flat extensions with charge

Alaee¹,

Pacheco²,

Cederbaum³

2019

Preprint

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The Bartnik mass is a notion of quasi-local mass which is remarkably difficult to compute. Mantoulidis and Schoen [14] developed a novel technique to construct asymptotically flat extensions of minimal Bartnik data in such a way that the ADM mass of these extensions is well-controlled, and thus, they were able to compute the Bartnik mass for minimal spheres satisfying a stability condition. In this work, we develop extensions and gluing tools, à la Mantoulidis-Schoen, for time-symmetric initial data sets for the Einstein-Maxwell equations that allow us to compute the value of an ad-hoc notion of charged Barnik mass for suitable charged minimal Bartnik data.

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“…Proof. First, from (2.5) and (2.6) with v ≡ 1 and E = u, we know that the scalar curvature of a metric of the form γ = ds 2 + u(s) 2 g * is well-known (see, for example, [7]) to be given by…”

Section: Gluing Via An R > τ Bridge In Spherical Symmetrymentioning

confidence: 99%

Asymptotically hyperbolic extensions and an analogue of the Bartnik mass

Pacheco

Cederbaum

McCormick

2018

Journal of Geometry and Physics

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The Bartnik mass is a quasi-local mass tailored to asymptotically flat Riemannian manifolds with non-negative scalar curvature. From the perspective of general relativity, these model time-symmetric domains obeying the dominant energy condition without a cosmological constant. There is a natural analogue of the Bartnik mass for asymptotically hyperbolic Riemannian manifolds with a negative lower bound on scalar curvature which model timesymmetric domains obeying the dominant energy condition in the presence of a negative cosmological constant.Following the ideas of Mantoulidis and Schoen [16], of Miao and Xie [20], and of joint work of Miao and the authors [6], we construct asymptotically hyperbolic extensions of minimal and constant mean curvature (CMC) Bartnik data while controlling the total mass of the extensions. We establish that for minimal surfaces satisfying a stability condition, the Bartnik mass is bounded above by the conjectured lower bound coming from the asymptotically hyperbolic Riemannian Penrose inequality. We also obtain estimates for such a hyperbolic Bartnik mass of CMC surfaces with positive Gaussian curvature.

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Higher dimensional black hole initial data with prescribed boundary metric

Cited by 18 publications

References 12 publications

Minimal hypersurfaces and boundary behavior of compact manifolds with nonnegative scalar curvature

Minimal hypersurfaces and boundary behavior of compact manifolds with nonnegative scalar curvature

Asymptotically flat extensions with charge

Asymptotically hyperbolic extensions and an analogue of the Bartnik mass

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