Effective versions of the positive mass theorem

The study of stable minimal surfaces in Riemannian 3-manifolds ( M , g ) with non-negative scalar curvature has a rich history. In this paper, we prove rigidity of such surfaces when ( M , g ) is asymptotically flat and has horizon boundary. As a consequence, we obtain an effective version of the positive mass theorem in terms of isoperimetric or, more generally, closed volume-preserving stable CMC surfaces that is appealing from both a physical and a purely geometric point of view. We also include a proof of the following conjecture of Schoen: An asymptotically flat Riemannian 3-manifold with non-negative scalar curvature that contains an unbounded area-minimizing surface is isometric to flat R 3 .