Mathematical Research Letters

Volume 25 (2018)

Number 3

Higher dimensional black hole initial data with prescribed boundary metric

Pages: 937 – 956

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

Authors

Armando J. Cabrera Pacheco (Department of Mathematics, University of Miami, Coral Gables, Florida, U.S.A.; and Department of Mathematics, Universität Tübingen, Germany)

Pengzi Miao (Department of Mathematics, University of Miami, Coral Gables, Florida, U.S.A.)

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.

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Received 11 August 2015