On the near-equality case of the Positive Mass Theorem

The Positive Mass Conjecture states that any complete asymptotically flat manifold of nonnnegative scalar curvature has nonnegative mass. Moreover, the equality case of the Positive Mass Conjecture states that in the above situation, if the mass is zero, then the Riemannian manifold must be Euclidean space. The Positive Mass Conjecture was proved by R. Schoen and S.-T. Yau for all manifolds of dimension less than 8 [SY79], and it was proved by E. Witten for all spin manifolds [Wit81]. In this paper, we consider complete asymptotically flat manifolds of nonnegative scalar curvature that are also harmonically flat in an end. We show that, whenever the Positive Mass Theorem holds, any appropriately normalized sequence of such manifolds whose masses converge to zero must have metrics that are uniformly converging to Euclidean metrics outside a compact region. This result is an ingredient in a forthcoming proof, co-authored with H. Bray, of the Riemannian Penrose inequality in dimensions less than 8 [BL].