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On the geometry and topology of initial data sets with horizons

Andersson, L., Dahl, M., Galloway, G. J., & Pollack, D. (2018). On the geometry and topology of initial data sets with horizons. The Asian journal of mathematics, 22(5), 863-882. doi:10.4310/AJM.2018.v22.n5.a4.

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Genre: Journal Article

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Creators:
Dahl, Mattias, Author
Galloway, Gregory J., Author
Pollack, Daniel, Author
Affiliations:
1Geometric Analysis and Gravitation, AEI-Golm, MPI for Gravitational Physics, Max Planck Society, ou_24012

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Free keywords: General Relativity and Quantum Cosmology, gr-qc,Mathematics, Differential Geometry, math.DG
Abstract: 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 $\partial M$ of $M$ is a collection of Marginally Outer (or Inner) Trapped Surfaces (MOTSs and/or MITSs) and such that $M\setminus \partial M$ 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 $3\leq n \leq 7$, 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 $\partial M$ induced by $g$. This result may be viewed as a generalization of Galloway and Schoen's higher dimensional black hole topology theorem 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 by Eichmair, Galloway, and Pollack in dimension $n=3$, to higher dimensions.

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Dates: 2015-08-0820152018
Publication Status: Published in print
Pages: -
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Identifiers: arXiv: 1508.01896
URI: http://arxiv.org/abs/1508.01896
DOI: 10.4310/AJM.2018.v22.n5.a4
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Title: The Asian journal of mathematics
Source Genre: Journal
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Publ. Info: Bethlehem, Pa.
Pages: - Volume / Issue: 22 (5) Sequence Number: - Start / End Page: 863 - 882 Identifier: -