Effective versions of the positive mass theorem

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 R 3 .

Asymptotically flat three-manifolds contain minimal planes

“…We note that the argument used here should extend (using e.g., arguments from [8]) to show equivalence of finite index and finite total curvature for embedded minimal surfaces in asymptotically flat 3-manifolds. 5 We note that in the case of ambient R 3 the equivalence of finite index and finite total curvautre is a well known result of Fischer-Colbrie [18].…”

A Splitting Theorem for Scalar Curvature

“…We refer to [23,17,3,20] as well as Appendix C of [8] for discussions and proofs of the following refinement of the latter alternative in Lemma 2.2.…”

“…These ideas have been developed by the first-and second-named authors to establish the following scalar-curvature rigidity result for asymptotically flat 3-manifolds which had been conjectured by R. Schoen. 8]). The only asymptotically flat Riemannian 3-manifold with non-negative scalar curvature that admits a non-compact, area-minimizing boundary is flat R 3 .…”

Characterization of Large Isoperimetric Regions in Asymptotically Hyperbolic Initial Data

“…The optimal, global uniqueness result for stable constant mean curvature spheres in initial data asymptotic to Schwarzschild has recently been established by the first-and the secondnamed authors in [8,9], building on earlier work of G. Huisken and S.-T. Yau [18], of J. Qing and G. Tian [24], of J. Metzger and the second-named author [14], of S. Brendle and the secondnamed author [4], as well as that of A. Carlotto and the first-and second-named authors [5]. We refer to the introduction of [8] for a comprehensive account and more detailed description of these and other important contributions in this context.…”