Rigidity of Stable Marginally Outer Trapped Surfaces in Initial Data Sets

Carlotto, Alessandro

doi:10.1007/s00023-016-0477-6

Cited by 4 publications

(7 citation statements)

References 36 publications

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“…First: as an immediate consequence of our gluing scheme we are able to produce data that are flat on a half-space and therefore contain plenty of stable (in fact: locally areaminimizing) minimal hypersurfaces, a conclusion which comes quite unexpected based on various recent scalar curvature rigidity results both in the closed and in the free-boundary case (see the works [BBN10,Nun13,MM15,Amb15]). As a result, combining this fact with with the rigidity counterparts obtained by the first-named author, contained in [Car13] and [Car14], we are able to provide a rather exhaustive answer to the fundamental problem of existence of stable minimal hypersurfaces in asymptotically flat manifolds (more generally: marginally outer trapped hypersurfaces in initial data sets). Furthermore, the reader shall notice that, again in the time-symmetric case, our solutions contain outlying volumepreserving stable constant mean curvature spheres that enclose arbitrarily large volumes.…”

Section: Introductionmentioning

confidence: 82%

Localizing solutions of the Einstein constraint equations

Carlotto

Schoen

2015

Invent. math.

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We perform an optimal localization of asymptotically flat initial data sets and construct data that have positive ADM mass but are exactly trivial outside a cone of arbitrarily small aperture. The gluing scheme that we develop allows to produce a new class of N -body solutions for the Einstein equation, which patently exhibit the phenomenon of gravitational shielding: for any large T we can engineer solutions where any two massive bodies do not interact at all for any time t ∈ (0, T ), in striking contrast with the Newtonian gravity scenario.

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Section: Introductionmentioning

confidence: 82%

Localizing solutions of the Einstein constraint equations

Carlotto

Schoen

2015

Invent. math.

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“…If equality holds in (3.19) then, by similar reasoning as before, one sees that µ + J(ν) = c and χ = 0. Finally, we mention that results concerning the infinitesimal rigidity of noncompact stable minimal MOTS have been obtained in [13].…”

Section: Infinitesimal Rigiditymentioning

confidence: 95%

Rigidity of marginally outer trapped $2$-spheres

Galloway

Mendes

2018

Communications in Analysis and Geometry

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In a matter-filled spacetime, perhaps with positive cosmological constant, a stable marginally outer trapped 2-sphere must satisfy a certain area inequality. Namely, as discussed in the paper, its area must be bounded above by 4π/c, where c > 0 is a lower bound on a natural energy-momentum term. We then consider the rigidity that results for stable, or weakly outermost, marginally outer trapped 2-spheres that achieve this upper bound on the area. In particular, we prove a splitting result for 3-dimensional initial data sets analogous to a result of Bray, Brendle and Neves [10] concerning area minimizing 2-spheres in Riemannian 3-manifolds with positive scalar curvature. We further show that these initial data sets locally embed as spacelike hypersurfaces into the Nariai spacetime. Connections to the Vaidya spacetime and dynamical horizons are also discussed.

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“…On the other hand, by [16], we can conclude the following. |P | is a stable MOTS on which h t + k| T Σt = 0 (spacetime totally geodesic) by (8.1), where h t is the second fundamental form of Σ t with respect to ∇u |∇u| .…”

Section: Proof Of Corollary 12mentioning

confidence: 68%

“…Besides, µ + J, ∇u |∇u| = 0 on Σ t . Furthermore by [16] Theorem 1 (2), we know that Σ t has vanishing Gauss curvature.…”

Section: Proof Of Corollary 12mentioning

confidence: 99%

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On a spacetime positive mass theorem with corners

Tin-Yau¹

2021

Preprint

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In this paper we consider the positive mass theorem for general initial data sets satisfying the dominant energy condition which are singular across a piecewise smooth surface. We find jump conditions on the metric and second fundamental form which are sufficient for the positivity of the total spacetime mass. Our method extends that of [30] to the singular case (which we refer to as initial data sets with corners) using some ideas from [31]. As such we give an integral lower bound on the spacetime mass and we characterise the case of zero mass. Our approach also leads to a new notion of quasilocal mass which we show to be positive, extending the work of [54] to the spacetime case. Moreover, we give sufficient conditions under which spacetime Bartnik data sets cannot admit a fill-in satisfying the dominant energy condition. This generalises the work of [58] and [57] to the spacetime setting.

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Rigidity of Stable Marginally Outer Trapped Surfaces in Initial Data Sets

Cited by 4 publications

References 36 publications

Localizing solutions of the Einstein constraint equations

Localizing solutions of the Einstein constraint equations

Rigidity of marginally outer trapped $2$-spheres

On a spacetime positive mass theorem with corners

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