MiS Preprint Repository

The Positive Mass Theorem implies in the case of equality (that is, when the total mass is zero) that any smooth, complete, asymptotically flat 3-manifold with non-negative scalar curvature which has zero total mass is isometric to $(R^3,{\delta }_{ij})$. In this paper, we quantify this statement using spinors and prove that if a complete, asymptotically flat manifold with non-negative scalar curvature has small mass and bounded isoperimetric constant, then the manifold must be close to $(R^3,{\delta }_{ij})$, in the sense that there is an upper bound for the $L^2$ norm of the Riemannian curvature tensor over the manifold except for a set of small measure. This curvature estimate allows us to extend the case of equality of the Positive Mass Theorem to include non-smooth manifolds with generalized non-negative scalar curvature, which we define.