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On the geometry and topology of initial data sets with horizons
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).ORCID iD: 0000-0002-9184-1467
2018 (English)In: Asian Journal of Mathematics, ISSN 1093-6106, E-ISSN 1945-0036, Vol. 22, no 5, p. 863-882Article in journal (Refereed) Published
Abstract [en]

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.

Place, publisher, year, edition, pages
International Press of Boston, Inc. , 2018. Vol. 22, no 5, p. 863-882
Keywords [en]
Initial data set, Jang's equation, Marginally outer trapped surfaces
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-246525DOI: 10.4310/AJM.2018.v22.n5.a4Scopus ID: 2-s2.0-85058986797OAI: oai:DiVA.org:kth-246525DiVA, id: diva2:1297483
Note

QC 20190320

Available from: 2019-03-19 Created: 2019-03-20 Last updated: 2019-03-20Bibliographically approved

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Dahl, Mattias

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