Geometric Stability of the Schoen-Yau Zero Mass Theorem

TL;DR

This paper reviews the geometric stability of the Schoen-Yau Zero Mass Theorem, exploring how nearly zero mass manifolds approximate Euclidean space and discussing various notions of convergence.

Contribution

It surveys existing results, examples, and open questions regarding the stability of the zero mass rigidity in the positive mass theorem.

Findings

Examples of manifolds with mass approaching zero

Various geometric notions of convergence analyzed

Open questions on the best convergence notion for stability

Abstract

In 1979, Schoen and Yau proved their famous Positive Mass Theorem which is a combination of a comparison theorem: {\em a three dimensional asymptotically flat Riemannian manifold with nonnegative scalar curvature has nonnegative ADM mass}, and a rigidity theorem: {\em if such a manifold has zero ADM mass then it is isometric to Euclidean space}. Here we review results and open questions on the geometric stability of their zero mass rigidity theorem: {\em if such a manifold has almost zero mass, how close is its geometry to that of Euclidean space}? We review the geometry of these spaces, examples of sequences of such spaces with mass approaching zero, and a variety of geometric notions of convergence. Although there has been much progress, it is still an open question (even in dimension three): exactly which geometric notion of convergence works best to capture the geometric stability of this famous rigidity theorem.