Blowup of Jang’s Equation at Outermost Marginally Trapped Surfaces

Metzger, Jan

doi:10.1007/s00220-009-0934-x

Cited by 22 publications

(37 citation statements)

References 9 publications

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“…In order to obtain the required boundary gradient estimate at this inner boundary, we must impose the ambient mean curvature restriction mentioned previously, that tr M (K) = g ij K ij is nonnegative on M \E 0 . Similarly, it was observed by J. Metzger [13] that restricting to tr M K ≥ 0 in the capillarity regularised problem prevents the solution from blowing-up to negative infinity over marginally inner trapped surfaces in the initial data set.…”

Section: Level-set Description and Elliptic Regularisationsupporting

confidence: 56%

On the evolution of hypersurfaces by their inverse null mean curvature

Moore¹

2014

J. Differential Geom.

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We introduce a new geometric evolution equation for hypersurfaces in asymptotically flat spacetime initial data sets, that unites the theory of marginally outer trapped surfaces (MOTS) with the study of inverse mean curvature flow in asymptotically flat Riemannian manifolds. A theory of weak solutions is developed using level-set methods and an appropriate variational principle. This new flow has a natural application as a variational-type approach to constructing MOTS, and this work also gives new insights into the theory of weak solutions of inverse mean curvature flow.

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Section: Level-set Description and Elliptic Regularisationsupporting

confidence: 56%

On the evolution of hypersurfaces by their inverse null mean curvature

Moore¹

2014

J. Differential Geom.

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“…From Theorem 2.2, Ω i+ = ∅ for all i. In fact, as in [25], the maximum principle implies that Ω i+ ⊃ V i := ∪ t∈[ 1 i , ] Σ t . This implies, in particular, that S i = ∅ for all i.…”

Section: )mentioning

confidence: 87%

“…and studies the limit as τ → 0. This regularized equation satisfies an a priori height estimate that allows one to construct a smooth global solution u τ on M \ Σ such that u τ → 0 on the asymptotically flat end (uniformly in τ ), and u τ → ∞ as τ → 0 on a fixed neighborhood of Σ; see [34, 3,7,25,12]. To get smooth convergence up to dimension 7, one applies the method of regularity introduced in the study of MOTSs by Eichmair [7], based on the C-minimizing property.…”

Section: )mentioning

confidence: 99%

On the geometry and topology of initial data sets with horizons

Andersson

Dahl

Galloway

et al. 2018

Asian Journal of Mathematics

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We study the relationship between initial data sets with horizons and the existence of metrics of positive scalar curvature. We define a Cauchy Domain of Outer Communications (CDOC) to be an asymptotically flat initial set (M, g, K) such that the boundary ∂M of M is a collection of Marginally Outer (or Inner) Trapped Surfaces (MOTSs and/or MITSs) and such that M \ ∂M contains no MOTSs or MITSs. This definition is meant to capture, on the level of the initial data sets, the well known notion of the domain of outer communications (DOC) as the region of spacetime outside of all the black holes (and white holes). Our main theorem establishes that in dimensions 3 ≤ n ≤ 7, a CDOC which satisfies the dominant energy condition and has a strictly stable boundary has a positive scalar curvature metric which smoothly compactifies the asymptotically flat end and is a Riemannian product metric near the boundary where the cross sectional metric is conformal to a small perturbation of the initial metric on the boundary ∂M induced by g. This result may be viewed as a generalization of Galloway and Schoen's higher dimensional black hole topology theorem [17] to the exterior of the horizon. We also show how this result leads to a number of topological restrictions on the CDOC, which allows one to also view this as an extension of the initial data topological censorship theorem, established in [10] in dimension n = 3, to higher dimensions.

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“…A chosen end can be separated from all the other ends of (M, g, k) by an outermost marginally trapped surface, see [3] for the case n = 3, and [14] for a different proof that works for all 3 ≤ n < 8. The main result in [21] (see alternatively [14, Remark 4.1]) shows that one can find a complete connected hypersurface Σ = graph(f Σ , U Σ ) as in (d) of Proposition 7 such that U Σ contains the exterior region of the chosen end, and such that U Σ lies beyond the outermost marginally trapped surface that separates the chosen end from the other ends. The rest of the proof can then proceed exactly as in the case of one end.…”

Section: Introductionmentioning

confidence: 99%