On a Penrose-Like Inequality in Dimensions Less than Eight

McCormick, Stephen; Miao, Pengzi

doi:10.1093/imrn/rnx181

Cited by 24 publications

(33 citation statements)

References 17 publications

(30 reference statements)

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“…Recall that if (M, g) is asymptotically flat with boundary ∂M, then we say ∂M is outward-minimizing if |S| ≥ |∂M| for all surfaces S enclosing ∂M, where | · | denotes the hypersurface area (with respect to g). The following theorem was recently proved by McCormick and Miao [17].…”

Section: Introductionmentioning

confidence: 89%

“…For k ≥ 2 and τ > 0, we let Met k −τ (M) denote the set of C k Riemannian metrics g on M satisfying (17) in the fixed coordinate chart. (The ADM mass of g ∈ Met k −τ (M) is well-defined if τ > n−2 2 and the scalar curvature of g is integrable [2,9].)…”

Section: Lower Semicontinuity For Weighted C 2 Convergence In All Dimmentioning

confidence: 99%

“…Generally, the missing link in establishing Theorem 1 in higher dimensions has been a useful quantitative lower bound for the ADM mass in terms of the geometry of an outward-minimizing surface. Fortunately, a recent result of S. McCormick and P. Miao [17] provides such an estimate (see Theorem 8 below) that is sufficient for our purposes. Their work uses the Riemannian Penrose inequality in higher dimensions, due to H. Bray and D. Lee [7] (which itself was a generalization of Bray's original proof in dimension three [6]).…”

Section: Introductionmentioning

confidence: 96%

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Lower semicontinuity of the ADM mass in dimensions two through seven

Jauregui

2019

Pacific J. Math.

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The semicontinuity phenomenon of the ADM mass under pointed (i.e., local) convergence of asymptotically flat metrics is of interest because of its connections to nonnegative scalar curvature, the positive mass theorem, and Bartnik's massminimization problem in general relativity. In this paper, we extend a previously known semicontinuity result in dimension three for C 2 pointed convergence to higher dimensions, up through seven, using recent work of S. McCormick and P. Miao (which itself builds on the Riemannian Penrose inequality of H. Bray and D. Lee). For a technical reason, we restrict to the case in which the limit space is asymptotically Schwarzschild. In a separate result, we show that semicontinuity holds under weighted, rather than pointed, C 2 convergence, in all dimensions n ≥ 3, with a simpler proof independent of the positive mass theorem. Finally, we also address the two-dimensional case for pointed convergence, in which the asymptotic cone angle assumes the role of the ADM mass.

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Section: Introductionmentioning

confidence: 89%

Section: Lower Semicontinuity For Weighted C 2 Convergence In All Dimmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 96%

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Lower semicontinuity of the ADM mass in dimensions two through seven

Jauregui

2019

Pacific J. Math.

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“…This may be considered as a generalization of the Penrose inequality with corners, established in [45,47], to the setting that includes asymptotically cylindrical ends. We may now apply Proposition 4.2 to each component of Σ * to obtain the desired inequality (2.6).…”

Section: Brown-york Mass Angular Momentum and Charge Inequalitiesmentioning

confidence: 99%

Geometric Inequalities for Quasi-Local Masses

Alaee

Khuri

Yau

2020

Commun. Math. Phys.

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In this paper lower bounds are obtained for quasi-local masses in terms of charge, angular momentum, and horizon area. In particular we treat three quasi-local masses based on a Hamiltonian approach, namely the Brown-York, Liu-Yau, and Wang-Yau masses. The geometric inequalities are motivated by analogous results for the ADM mass. They may be interpreted as localized versions of these inequalities, and are also closely tied to the conjectured Bekenstein bounds for entropy of macroscopic bodies. In addition, we give a new proof of the positivity property for the Wang-Yau mass which is used to remove the spin condition in higher dimensions. Furthermore, we generalize a recent result of Lu and Miao to obtain a localized version of the Penrose inequality for the static Wang-Yau mass.

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“…But (3.6) violates the Riemannian Penrose inequality (for metrics possibly with corner along a hypersurface, cf. [11]). Thus, we must have…”

Section: Equality Case Of the Localized Penrose Inequalitymentioning

confidence: 99%

Variation and rigidity of quasi-local mass

Miao

2019

Advances in Theoretical and Mathematical Physics

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Inspired by the work of Chen-Zhang [5], we derive an evolution formula for the Wang-Yau quasi-local energy in reference to a static space, introduced by Chen-Wang-Wang-Yau [4]. If the reference static space represents a mass minimizing, static extension of the initial surface Σ, we observe that the derivative of the Wang-Yau quasi-local energy is equal to the derivative of the Bartnik quasi-local mass at Σ.Combining the evolution formula for the quasi-local energy with a localized Penrose inequality proved in [10], we prove a rigidity theorem for compact 3-manifolds with nonnegative scalar curvature, with boundary. This rigidity theorem in turn gives a characterization of the equality case of the localized Penrose inequality in 3-dimension.

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On a Penrose-Like Inequality in Dimensions Less than Eight

Cited by 24 publications

References 17 publications

Lower semicontinuity of the ADM mass in dimensions two through seven

Lower semicontinuity of the ADM mass in dimensions two through seven

Geometric Inequalities for Quasi-Local Masses

Variation and rigidity of quasi-local mass

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