Abstract
A spatial Schwarzschild manifold of mass m is an asymptotically flat manifold that represents the {t=0} spacelike slice of the Schwarzschild spacetime, a spacetime that models the gravitational field surrounding a spherically symmetric non-rotating massive body. The Riemannian Penrose inequality establishes an upper bound for the ADM mass of an asymptotically flat manifold with non-negative scalar curvature in terms of the volume of its outermost apparent horizon; remarkably, this inequality is rigid and equality is achieved for Schwarzschild manifolds. Our main results are that any metric g of positive scalar curvature on the 3-sphere can be realized as the induced metric of 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 determined by the Riemannian Penrose inequality. Along the same lines, any metric g of positive scalar curvature on the n-sphere, with n greater or equal than 4, such that it isometrically embeds into the (n+1)-Euclidean space as a star-shaped surface, 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. In addition to the extension problems above, motivated by various definitions of quasi-local masses, we study an isometric embedding problem of 2-spheres into Schwarzschild manifolds. We prove that if g is a Riemannian metric on the 2-sphere which is close to the standard metric, then it admits an isometric embedding into any spatial Schwarzschild manifold with small mass.