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

Lu, Siyuan; Miao, Pengzi

doi:10.48550/arxiv.1703.08164

Cited by 13 publications

(18 citation statements)

References 31 publications

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“…Proof. This result follows by combining the proof of Theorem 2.8 above, and the proof of Theorem 1.1 in [43]. Here we provide an outline.…”

Section: Penrose-like Inequality For Quasi-local Mass With a Static R...mentioning

confidence: 77%

“…where H is the mean curvature of the embedding Σ ֒→ Ω. Recently, Lu and Miao [43] have proven a Penrose type inequality for the static Brown-York mass in which the reference static spacetime is the Schwarzschild solution. Here we give an extension of their result to the static Liu-Yau mass setting.…”

Section: Penrose-like Inequality For Quasi-local Mass With a Static R...mentioning

confidence: 99%

“…Here we show how to smooth the gluing construction to allow for a conformal change to nonnegative scalar curvature, and thus avoid having to use spinors to establish positive mass. Lastly, we will also generalize a result of Lu-Miao [43] to obtain a localized version of the Penrose inequality for the static Wang-Yau mass [15]. In the remainder of the paper geometrized units will be used so that c = G = 1.…”

mentioning

confidence: 97%

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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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“…Proof. This result follows by combining the proof of Theorem 2.8 above, and the proof of Theorem 1.1 in [43]. Here we provide an outline.…”

Section: Penrose-like Inequality For Quasi-local Mass With a Static R...mentioning

confidence: 77%

Section: Penrose-like Inequality For Quasi-local Mass With a Static R...mentioning

confidence: 99%

mentioning

confidence: 97%

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Geometric Inequalities for Quasi-Local Masses

Alaee

Khuri

Yau

2020

Commun. Math. Phys.

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“…Remark 2.1. A monotonicity generalizing (2.17) with the background Schwarzschild metric replaced by a general static metric can be found in [12].…”

Section: Proposition 22 ([16]mentioning

confidence: 99%

Bartnik mass via vacuum extensions

Miao

Xie

2019

Preprint

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We construct asymptotically flat, scalar flat extensions of Bartnik data (Σ, γ, H), where γ is a metric of positive Gauss curvature on a two-sphere Σ, and H is a function that is either positive or identically zero on Σ, such that the mass of the extension can be made arbitrarily close to the half area radius of (Σ, γ).In the case of H ≡ 0, the result gives an analogue of a theorem of Mantoulidis and Schoen [13], but with extensions that have vanishing scalar curvature. In the context of initial data sets in general relativity, the result produces asymptotically flat, time-symmetric, vacuum initial data with an apparent horizon (Σ, γ), for any metric γ with positive Gauss curvature, such that the mass of the initial data is arbitrarily close to the optimal value in the Riemannian Penrose inequality.The method we use is the Shi-Tam type metric construction from [19] and a refined Shi-Tam monotonicity, found by the first named author in [16].

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“…Their mass was later on generalized and further studied by Wang and Yau, see [WY07,WY09]. On the other hand, Lu and Miao derived a quasi-local mass type inequality in the Schwarzschild space, see [LM17].…”

Section: ∂Mmentioning

confidence: 99%

A free boundary isometric embedding problem in the unit ball

Koerber¹

2019

Preprint

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In this article, we study a free boundary isometric embedding problem for abstract Riemannian two-manifolds with the topology of the disc. Under the assumption of positive Gauss curvature and geodesic curvature of the boundary being equal to one, we show that any such disc may be isometrically embedded into the Euclidean three space R 3 such that the image of the boundary meets the unit sphere S 2 orthogonally. Moreover, we also show that the embedding is unique up to rotations and reflections through planes containing the origin. Finally, we define a new Brown-York type quasi-local mass for certain free boundary surfaces and discuss its positivity. * is non-empty but we would of course like to show that G k,α * = G k,α . In order to do so, we equip G k,α with the C k,α (D, Sym(R 2 ))−topology and show that G k,α is path-connected, open and closed. It actually turns out that G k,α is not only path-connected but that the paths can be chosen to be analytic maps from the unit interval to G k,α .

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

Cited by 13 publications

References 31 publications

Geometric Inequalities for Quasi-Local Masses

Geometric Inequalities for Quasi-Local Masses

Bartnik mass via vacuum extensions

A free boundary isometric embedding problem in the unit ball

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