Stable surfaces and free boundary marginally outer trapped surfaces

Alaee, Aghil; Lesourd, Martin; Yau, Shing–Tung

doi:10.1007/s00526-021-02063-w

Cited by 9 publications

(13 citation statements)

References 39 publications

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“…they admit metrics of positive scalar curvature. Their result is consistent with the example presented by R. Emparan and H. S. Reall [10] of a (4 + 1)-dimensional stationary vacuum black hole spacetime with horizon topology S 2 × S 1 .…”

Section: Introductionsupporting

confidence: 91%

“…The proof of Lemma 3.1 is essentially contained in the proof of Proposition 5.4 in [1]. But, for the sake of completeness, we include it here.…”

Section: Auxiliary Resultsmentioning

confidence: 99%

“…It is important to note that, because L is not necessarily symmetric and Σ is a manifold with boundary, there is no standard general existence result for the principal eigenvalue λ 1 (L) of L (with Robin boundary condition Bφ = 0) as in the closed case. Then, in order to guarantee the existence of λ 1 (L) with similar properties to those of the closed case, Alaee, Lesourd, and Yau [1] assumed that II ∂M (N, N) ≤ 0 along ∂Σ. But here, we succeed in circumventing this technical issue and do not make such an assumption.…”

Section: Auxiliary Resultsmentioning

confidence: 99%

“…In analogy with the minimal surface case, a closed MOTS Σ is said to be stable provided λ 1 (L) ≥ 0. Now, we are going to present a notion of stability for capillary MOTS in initial data sets with boundary introduced by A. Alaee, M. Lesourd, and S.-T. Yau [1]. We restrict our attention to the free boundary case.…”

Section: Marginally Outer Trapped Surfacesmentioning

confidence: 99%

“…where φ = V, N (see [1,Section 5]). The corresponding eigenvalue problem to functional (3) is the following:…”

Section: Marginally Outer Trapped Surfacesmentioning

confidence: 99%

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Rigidity of free boundary MOTS

Mendes¹

2021

Preprint

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The aim of this work is to present an initial data version of Hawking's theorem on the topology of back hole spacetimes in the context of manifolds with boundary. More precisely, we generalize the results of G. J. Galloway and R. Schoen [13] and G. J. Galloway [11, 12] by proving that a compact free boundary stable marginally outer trapped surface (MOTS) Σ in an initial data set with boundary satisfying natural dominant energy conditions (DEC) is of positive Yamabe type, i.e. Σ admits a metric of positive scalar curvature with minimal boundary, provided Σ is outermost. To do so, we prove that if Σ is a compact free boundary stable MOTS which does not admit a metric of positive scalar curvature with minimal boundary in an initial data set satisfying the interior and the boundary DEC, then an outer neighborhood of Σ can be foliated by free boundary MOTS Σ t , assuming that Σ is weakly outermost. Moreover, each Σ t has vanishing outward null second fundamental form, is Ricci flat with totally geodesic boundary, and the dominant energy conditions saturate on Σ t .

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Section: Introductionsupporting

confidence: 91%

“…The proof of Lemma 3.1 is essentially contained in the proof of Proposition 5.4 in [1]. But, for the sake of completeness, we include it here.…”

Section: Auxiliary Resultsmentioning

confidence: 99%

Section: Auxiliary Resultsmentioning

confidence: 99%

Section: Marginally Outer Trapped Surfacesmentioning

confidence: 99%

“…where φ = V, N (see [1,Section 5]). The corresponding eigenvalue problem to functional (3) is the following:…”

Section: Marginally Outer Trapped Surfacesmentioning