The Plateau problem for marginally outer trapped surfaces

Eichmair, Michael

doi:10.4310/jdg/1264601035

Cited by 69 publications

(101 citation statements)

References 39 publications

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“…Furthermore, the works of Andersson, Eichmair, and Metzger [2,10,1] imply that with out loss of generality, the above conjecture may be reduced to a simpler case. Define an outermost time-independent apparent horizon to be a time-independent apparent horizon which is not enclosed by any other.…”

Section: Time-independent Apparent Horizonsmentioning

confidence: 99%

“…Define an outermost time-independent apparent horizon to be a time-independent apparent horizon which is not enclosed by any other. The works of Andersson, Eichmair, and Metzger [2,10,1] imply that given any time-independent outer trapped surface, there always exists an outermost time-independent apparent horizon which encloses it (though not necessarily uniquely).…”

Section: Time-independent Apparent Horizonsmentioning

confidence: 99%

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P. D. E.'s Which Imply the Penrose Conjecture

Bray¹,

Khuri²

2011

Asian Journal of Mathematics

130

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Abstract. In this paper, we show how to reduce the Penrose conjecture to the known Riemannian Penrose inequality case whenever certain geometrically motivated systems of equations can be solved. Whether or not these special systems of equations have general existence theories is therefore an important open problem. The key tool in our method is the derivation of a new identity which we call the generalized Schoen-Yau identity, which is of independent interest. Using a generalized Jang equation, we use this identity to propose canonical embeddings of Cauchy data into corresponding static spacetimes. In addition, we discuss the Carrasco-Mars counterexample to the Penrose conjecture for generalized apparent horizons (added since the first version of this paper was posted on the arXiv) and instead conjecture the Penrose inequality for time-independent apparent horizons, which we define.

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Section: Time-independent Apparent Horizonsmentioning

confidence: 99%

Section: Time-independent Apparent Horizonsmentioning

confidence: 99%

P. D. E.'s Which Imply the Penrose Conjecture

Bray¹,

Khuri²

2011

Asian Journal of Mathematics

130

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“…Inequality (2.27) is a Penrose-like inequality for the Liu-Yau energy, whereas (2.28) is akin to a localized version of the ADM mass-angular momentum-charge inequality which has so far only been established in the maximal case [16,18,39,55]. Furthermore, inequality (2.28) holds without the assumption of axisymmetry if the angular momentum contribution from the right-hand side is dropped, while the stability hypothesis can be removed if Σ h has only one component [2,22]. The constant γ is invariant under rescalings of the metric and hence independent of |Σ h |.…”

Section: Statement Of Main Resultsmentioning

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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“…More generally, Schoen and Yau's analysis shows that any solution arising from a limit of suitably regularized boundary-value problems can only blow up on an apparent horizon (see Section 3.5 of [6]). This was used in [7] and [8] to prove the existence of apparent horizons in the presence of suitable geometric barriers. (A simple example is when there exists a bounded set Ω ⊂ M with at least two boundary components and H ∂Ω − |tr ∂Ω h| > 0.…”

Section: Theorem 1 ([2]mentioning

confidence: 99%

Blowup solutions of Jang’s equation near a spacetime singularity

Aazami¹,

Cox²

2014

Class. Quantum Grav.

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We study Jang's equation on a one-parameter family of asymptotically flat, spherically symmetric Cauchy hypersurfaces in the maximally extended Schwarzschild spacetime. The hypersurfaces contain apparent horizons and are parametrized by their proximity to the singularity at r = 0. We show that on those hypersurfaces sufficiently close to the singularity, every radial solution to Jang's equation blows up. The proof depends only on the geometry in an arbitrarily small neighborhood of the singularity, suggesting that Jang's equation is in fact detecting the singularity. We comment on possible applications to the weak cosmic censorship conjecture.

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The Plateau problem for marginally outer trapped surfaces

Cited by 69 publications

References 39 publications

P. D. E.'s Which Imply the Penrose Conjecture

P. D. E.'s Which Imply the Penrose Conjecture

Geometric Inequalities for Quasi-Local Masses

Blowup solutions of Jang’s equation near a spacetime singularity

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