Abstract
<p>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 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 >= 4, such that (S-n, 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.</p>