Stability of Euclidean 3-space for the positive mass theorem

We show that the Euclidean 3-space $\mathbb{R}^3$ is stable for the Positive Mass Theorem in the following sense. Let $(M_i,g_i)$ be a sequence of complete asymptotically flat $3$-manifolds with nonnegative scalar curvature and suppose that the ADM mass $m(g_i)$ of one end of $M_i$ converges to $0$. Then for all $i$, there is a subset $Z_i$ in $M_i$ such that $M_i\setminus Z_i$ contains the given end, the area of the boundary $\partial Z_i$ converges to zero, and $(M_i\setminus Z_i,g_i)$ converges to $\mathbb{R}^3$ in the pointed measured Gromov-Hausdorff topology for any choice of basepoints. This confirms a conjecture of G. Huisken and T. Ilmanen. Additionally, we find an almost quadratic upper bound for the area of $\partial Z_i$ in terms of $m(g_i)$. As an application of the main result, we also prove R. Bartnik's strict positivity conjecture.