Effective versions of the positive mass theorem

TL;DR

This paper establishes an effective geometric version of the positive mass theorem for asymptotically flat 3-manifolds with horizon boundary, using stable CMC surfaces, and proves a conjecture relating unbounded area-minimizing surfaces to flat space.

Contribution

It provides a new effective formulation of the positive mass theorem and proves a conjecture linking unbounded minimal surfaces to flat Euclidean space.

Findings

Rigidity of stable minimal surfaces in asymptotically flat manifolds.

An effective positive mass theorem using stable CMC surfaces.

Proof of Schoen's conjecture on unbounded area-minimizing surfaces.

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 $\mathbb{R}^3$.