A Penrose-like Inequality for the Mass of Riemannian Asymptotically Flat Manifolds

We prove an optimal Penrose-like inequality for the mass of any asymptotically flat Riemannian 3-manifold having an inner minimal 2-sphere and nonnegative scalar curvature. Our result shows that the mass is bounded from below by an expression involving the area of the minimal sphere (as in the original Penrose conjecture) and some nomalized Sobolev ratio. As expected, the equality case is achieved if and only if the metric is that of a standard spacelike slice in the Schwarzschild space.