Large-sphere and small-sphere limits of the Brown-York mass

Fan, Xu-Qian; Shi, Yuguang; Tam, Luen-Fai

doi:10.4310/cag.2009.v17.n1.a3

Cited by 77 publications

(76 citation statements)

References 18 publications

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“…Here the center-of-mass integral is also given in terms of a charge integral of the curvature. The large sphere limit of the energy for metrics with the weakest possible falloff conditions is calculated in [181, 462]. A further demonstration that the spatial infinity limit of the Brown-York energy in an asymptotically Schwarzschild spacetime is the ADM energy is given in [180].…”

Section: The Hamilton-jacobi Methodsmentioning

confidence: 99%

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Quasi-Local Energy-Momentum and Angular Momentum in General Relativity

Szabados

2009

Living Rev. Relativ.

324

390

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The present status of the quasi-local mass, energy-momentum and angular-momentum constructions in general relativity is reviewed. First, the general ideas, concepts, and strategies, as well as the necessary tools to construct and analyze the quasi-local quantities, are recalled. Then, the various specific constructions and their properties (both successes and deficiencies are discussed. Finally, some of the (actual and potential) applications of the quasi-local concepts and specific constructions are briefly mentioned.

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Section: The Hamilton-jacobi Methodsmentioning

confidence: 99%

“…On the other hand, the special light cone reference used in [118, 335] reproduces the expected result in nonvacuum, and yields [1/(90 G )] r 5 T abcd t a t b t c t d in vacuum. The small sphere limit was also calculated in [181] for small geodesic spheres in a time symmetric spacelike hypersurface.…”

Section: The Hamilton-jacobi Methodsmentioning

confidence: 99%

Quasi-Local Energy-Momentum and Angular Momentum in General Relativity

Szabados

2009

Living Rev. Relativ.

324

390

View full text Add to dashboard Cite

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“…Let g r be the induced metric on S r and letg r = r −2 g r . Identifying S r with S 2 = { |y| = 1} via a map y = r −1 x, one can deduce from (6.3) that g r − σ C 3 ≤ Cr −τ (6.4) (see (2.17) in [18] for instance). Here σ is the standard metric on S 2 with area 4π and C > 0 is a constant independent on r.…”

Section: Limits Along Isomeric Embeddings Of Large Spheres Into Schwamentioning

confidence: 99%

“…By (6.9), Let V (r) be the volume of the region enclosed by S r in (M,g). By (2.28) in [18], (6.24) V (r) = 1 2 rA(r) − 2 3 πr 3 + 2πmr 2 + o(r 2 ).…”

Section: 2mentioning

confidence: 99%

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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“…where H g 0 denotes the mean curvature of Σ c when embedded, with its induced metric γ c , into (R 3 , g 0 ). The right hand side is the Brown-York mass of Σ c and, by [35,12], it converges to m ADM (M, g) as c → 1. This recovers Bray's capacity inequality (1.2).…”

Section: )mentioning

confidence: 99%

Capacity, quasi-local mass, and singular fill-ins

Mantoulidis

Miao

Tam

2019

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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We derive new inequalities between the boundary capacity of an asymptotically flat 3-manifold with nonnegative scalar curvature and boundary quantities that relate to quasi-local mass; one relates to Brown-York mass and the other is new. We argue by recasting the setup to the study of mean-convex fill-ins with nonnegative scalar curvature and, in the process, we consider fillins with singular metrics, which may have independent interest. Among other things, our work yields new variational characterizations of Riemannian Schwarzschild manifolds and new comparison results for surfaces in them.

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Large-sphere and small-sphere limits of the Brown-York mass

Cited by 77 publications

References 18 publications

Quasi-Local Energy-Momentum and Angular Momentum in General Relativity

Quasi-Local Energy-Momentum and Angular Momentum in General Relativity

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

Capacity, quasi-local mass, and singular fill-ins

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