Geometric Stability of the Schoen-Yau Zero Mass Theorem

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.