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# Mathematical Research Letters

## Volume 25 (2018)

### Number 3

### Higher dimensional black hole initial data with prescribed boundary metric

Pages: 937 – 956

DOI: https://dx.doi.org/10.4310/MRL.2018.v25.n3.a10

#### Authors

#### Abstract

We obtain higher dimensional analogues of the results of Mantoulidis and Schoen. More precisely, we show that (i) any metric $g$ with positive scalar curvature on the $3$-sphere $S^3$ can be realized as the induced metric on the outermost apparent horizon of a $4$-dimensional asymptotically flat manifold with non-negative scalar curvature, whose ADM mass can be arranged to be arbitrarily close to the optimal value specified by the Riemannian Penrose inequality; (ii) any metric g with positive scalar curvature on the $n$-sphere $S^n$, with $n \geq 4$, such that $(S^n, g)$ isometrically embeds into $\mathbb{R}^{n+1}$ as a star-shaped hypersurface, can be realized as the induced metric on the outermost apparent horizon of an $(n + 1)$-dimensional asymptotically flat manifold with non-negative scalar curvature, whose ADM mass can be made to be arbitrarily close to the optimal value.

Received 11 August 2015

Accepted 11 May 2017

Published 3 August 2018