If Ricci is bounded below, then mass is in control!
Abstract: The ADM mass of an isolated gravitational system is a geometric invariant measuring the total mass due to matter and other fields. In a previous work, we showed how to compute this invariant (in 3 spatial dimensions) by studying harmonic functions. Now I'll use this formula to consider the following question: How flat is an asymptotically flat manifold with very little total mass? We make progress on this problem and confirm special cases of conjectures made by Huisken-Ilmanen and Sormani. The main results asserts that in the class of reasonably-behaved asymptotically flat manifolds with non-negative scalar curvature satisfying a uniform lower bound on Ricci curvature, small mass implies Gromov-Hausdorff closeness to flat space.