Higher dimensional black hole initial data with prescribed boundary metric

We obtain higher dimensional analogues of the results of Mantoulidis and Schoen in [8]. More precisely, we show that (i) any metric g with positive scalar curvature on the $3$-sphere S³ 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 Sⁿ, with n ≥ 4, such that (Sⁿ, g) isometrically embeds into Rⁿ⁺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.