TY - JOUR
T1 - Effective versions of the positive mass theorem
AU - Carlotto, Alessandro
AU - Chodosh, Otis
AU - Eichmair, Michael
N1 - Funding Information:
The first-named author wishes to express his gratitude to Richard Schoen for introducing him, with great professionality and unparalleled enthusiasm, to the mathematical challenges of general relativity. He also thankfully acknowledges the support of André Neves through his ERC Starting Grant. The second-named author would like to convey his deepest thanks to his advisor Simon Brendle for his invaluable support and continued encouragement. His research was supported in part by an NSF fellowship DGE-1147470 as well as the EPSRC grant EP/K00865X/1. The third-named author expresses his gratitude to Hubert Bray, Simon Brendle, Gregory Galloway, Gerhard Huisken, Jan Metzger, and Richard Schoen. A part of this paper was written up during his invigorating stay of two months at Stanford University, which was supported by their Mathematical Sciences Research Center. The second- and third-named authors would also like to thank the Erwin-Schrödinger-Institute of the University of Vienna for its hospitality during the special program "Dynamics of General Relativity: Numerical and Analytic Approaches" in the summer of 2011. Open access funding for this article was provided by University of Vienna. It is a pleasure to sincerely congratulate Richard Schoen on the occasion of his 65th birthday.
Funding Information:
The first-named author wishes to express his gratitude to Richard Schoen for introducing him, with great professionality and unparalleled enthusiasm, to the mathematical challenges of general relativity. He also thankfully acknowledges the support of André Neves through his ERC Starting Grant. The second-named author would like to convey his deepest thanks to his advisor Simon Brendle for his invaluable support and continued encouragement. His research was supported in part by an NSF fellowship DGE-1147470 as well as the EPSRC grant EP/K00865X/1. The third-named author expresses his gratitude to Hubert Bray, Simon Brendle, Gregory Galloway, Gerhard Huisken, Jan Metzger, and Richard Schoen. A part of this paper was written up during his invigorating stay of two months at Stanford University, which was supported by their Mathematical Sciences Research Center. The second- and third-named authors would also like to thank the Erwin-Schrödinger-Institute of the University of Vienna for its hospitality during the special program “Dynamics of General Relativity: Numerical and Analytic Approaches” in the summer of 2011. Open access funding for this article was provided by University of Vienna. It is a pleasure to sincerely congratulate Richard Schoen on the occasion of his 65th birthday.
Publisher Copyright:
© 2016, The Author(s).
PY - 2016/12/1
Y1 - 2016/12/1
N2 - 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.
AB - 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.
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U2 - 10.1007/s00222-016-0667-3
DO - 10.1007/s00222-016-0667-3
M3 - Article
AN - SCOPUS:84973137970
SN - 0020-9910
VL - 206
SP - 975
EP - 1016
JO - Inventiones Mathematicae
JF - Inventiones Mathematicae
IS - 3
ER -