Invent. math. DOI 10.1007/s00222-016-0667-3 Effective versions of the positive mass theorem Alessandro Carlotto1 · Otis Chodosh2 · Michael Eichmair3 Received: 2 March 2016 / Accepted: 24 April 2016 © The Author(s) 2016. This article is published with open access at Springerlink.com Abstract 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 R3. 1 Introduction The geometry of stable minimal and volume-preserving stable constant mean curvature surfaces in asymptotically flat 3-manifolds (M, g) with non-negative scalar curvature is witness to the physical properties of the space- times containing such (M, g) as maximal Cauchy hypersurfaces; see e.g. [9,10,20,37,38,55,64]. The transition from positive to non-negative scalar B Michael Eichmair
[email protected] 1 ETH Institute for Theoretical Studies, Clausiusstrasse 47, 8092 Zurich, Switzerland 2 DPMMS, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, UK 3 Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria 123 A.