Area-minimizing surfaces in asymptotically flat three-manifolds

TL;DR

This paper proves that an asymptotically flat three-manifold with non-negative scalar curvature must be flat Euclidean space if it contains an unbounded area-minimizing surface, answering a question posed by R. Schoen.

Contribution

It establishes a rigidity result linking the existence of unbounded area-minimizing surfaces to the flatness of the manifold, resolving a longstanding question.

Findings

Asymptotically flat manifolds with non-negative scalar curvature are flat if they contain unbounded area-minimizing surfaces.

The result characterizes the geometric structure of such manifolds, showing they are isometric to Euclidean space.

Provides a new criterion for flatness based on minimal surface properties.

Abstract

We show that an asymptotically flat Riemannian three-manifold with non-negative scalar curvature is isometric to flat $\mathbb{R}^3$ if it admits an unbounded area-minimizing surface. This answers a question of R. Schoen.