Positivity of Brown–York mass with quasi-positive boundary data

Shi, Yuguang; Tam, Luen-Fai

doi:10.4310/pamq.2019.v15.n3.a8

Cited by 4 publications

(3 citation statements)

References 15 publications

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“…From the results in [24,25], we see that m BY (∂Ω, g) is just the same as the original definition in [11,12] when the Gaussian curvature of the boundary (∂Ω, g) is nonnegative. As a corollary of Theorem 2.2, we have: Corollary 1.2.…”

Section: Introductionmentioning

confidence: 68%

Total mean curvature of the boundary and nonnegative scalar curvature fill-ins

Shi¹,

Wang²,

Wei³

2020

Preprint

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In the first part of the paper, we get some estimates for the supremum of the total mean curvature of boundaries of domains with nonnegative scalar curvature, and discuss its relationship with the positive mass theorem of asymptotically flat (hyperbolic) manifolds. In the second part of the paper, we prove the extensibility of an arbitrary boundary metric to a positive scalar curvature (PSC) metric inside for a compact manifold with nonempty boundaries which completely solve an open problem due to Gromov (see Question 1.1). The results in this paper also provide some partially affirmative answers to Gromov's conjectures formulated in [15] (see Conjecture 1.1, Conjecture 1.2 below).

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Section: Introductionmentioning

confidence: 68%

Total mean curvature of the boundary and nonnegative scalar curvature fill-ins

Shi¹,

Wang²,

Wei³

2020

Preprint

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“…Moreover, equality holds if and only if (Ω, g) is isometric to a domain in Euclidean space R 3 . Recently they extend the proof to quasi positive Gauss curvature [58], this means that K σ is nonnegative and is positive somewhere. In [ (2.4), Σ h is strictly stable and strictly outerminimizing, for any point in Ω its minimal geodesic line to Σ h is contained in Ω and its distance function to Σ h is smooth, then Ω is a domain in the canonical slice of a Schwarzschild spacetime.…”

Section: Statement Of Main Resultsmentioning

confidence: 98%

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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“…On the other hand, in [24] (also see an improvement in [26]), the first author and his collaborator proved the positivity of Brown-York mass introduced by Brown and York ( [4,5]).…”

Section: Introductionmentioning

confidence: 99%

On the Fill-in of Nonnegative Scalar Curvature Metrics

Shi

Wang

Wei

et al. 2019

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In the first part of this paper, we consider the problem of fill-in of nonnegative scalar curvature (NNSC) metrics for a triple of Bartnik data (Σ, γ, H). We prove that given a metric γ on S n−1 (3 ≤ n ≤ 7), (S n−1 , γ, H) admits no fill-in of NNSC metrics provided the prescribed mean curvature H is large enough (Theorem 1.4). Moreover, we prove that if γ is a positive scalar curvature (PSC) metric isotopic to the standard metric on S n−1 , then the much weaker condition that the total mean curvature ´Sn−1 H dµ γ is large enough rules out NNSC fill-ins, giving an partially affirmative answer to a conjecture by Gromov (see P. 23 in [12]). In the second part of this paper, we investigate the θ-invariant of Bartnik data and obtain some sufficient conditions for the existence of PSC fill-ins.

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Positivity of Brown–York mass with quasi-positive boundary data

Abstract: In this short note, we prove positivity of Brown-York mass under quasi-positive boundary data which generalize some previous results by the authors. The corresponding rigidity result is obtained.

Cited by 4 publications

References 15 publications

Total mean curvature of the boundary and nonnegative scalar curvature fill-ins

Total mean curvature of the boundary and nonnegative scalar curvature fill-ins

Geometric Inequalities for Quasi-Local Masses

On the Fill-in of Nonnegative Scalar Curvature Metrics

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