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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.