  We establish Gromov-Hausdorff stability of the Riemannian positive mass
theorem under the assumption of a Ricci curvature lower bound. More precisely,
consider a class of orientable complete uniformly asymptotically flat
Riemannian 3-manifolds with nonnegative scalar curvature, vanishing second
homology, and a uniform lower bound on Ricci curvature. We prove that if a
sequence of such manifolds has ADM mass approaching zero, then it must converge
to Euclidean 3-space in the pointed Gromov-Hausdorff sense. In particular, this
confirms Huisken and Ilmanen's conjecture on stability of the positive mass
theorem under the assumptions described above. The proof is based on the
harmonic level set approach to proving the positive mass theorem, combined with
techniques used in the proof of Cheeger and Colding's almost splitting theorem.
Furthermore, we show that the same results hold under a more general lower
bound on scalar curvature.
