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 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.