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 R. Schoen: An asymptotically flat Riemannian $3$-manifold with non-negative scalar curvature that contains an unbounded area-minimizing surface is isometric to flat ${\mathbb{R}}^3$.

All comments welcome! The final version has appeared in Invent. Math

References in corpus (5)

• Large isoperimetric regions in asymptotically hyperbolic manifolds
• A Compactness Theorem for the Second Fundamental Form
• Rigidity of area-minimizing two-spheres in three-manifolds
• Geometric relations of stable minimal surfaces and applications
• Variational and rigidity properties of static potentials

Cited by in corpus (21)

• Minimal hypersurfaces with bounded index
• On the existence of isoperimetric regions in manifolds with nonnegative Ricci curvature and Euclidean volume growth
• Large isoperimetric regions in asymptotically hyperbolic manifolds
• On far-outlying CMC spheres in asymptotically flat Riemannian $3$-manifolds
• Embeddings, immersions and the Bartnik quasi-local mass conjectures
• Isoperimetry, scalar curvature, and mass in asymptotically flat Riemannian $3$-manifolds
• The isoperimetric problem via direct method in noncompact metric measure spaces with lower Ricci bounds
• Asymptotically flat three-manifolds contain minimal planes
• A splitting theorem for scalar curvature
• Min-max minimal surfaces, horizons and electrostatic systems
• Rigidity of Hawking mass for surfaces in three manifolds
• Dihedral rigidity of parabolic polyhedrons in hyperbolic spaces
• A non-existence result for minimal catenoids in asymptotically flat spaces
• Characterization of large isoperimetric regions in asymptotically hyperbolic initial data
• The Willmore center of mass of initial data sets
• Doubling of asymptotically flat half-spaces and the Riemannian Penrose inequality
• Unstable CMC spheres and outlying CMC spheres in AF 3-manifolds
• A splitting theorem for manifolds with spectral nonnegative Ricci curvature and mean-convex boundary
• Spectral comparison and splitting theorems for the infinity-Bakry-Emery Ricci curvature
• Proper free-boundary minimal hypersurfaces with a rotational symmetry in the Schwarzschild space
• On the Positive Energy Theorem for Stationary Solutions to Fourth-Order Gravity