Lee–Sormani intrinsic flat stability conjecture for the positive mass theorem

Let $M_i$ be a sequence of asymptotically flat $3$-manifolds with nonnegative scalar curvature and ADM mass tending to zero. Let $M_{i,\mathrm{ext}}$ denote the exterior region, let $\Omega_i\subset M_i$ be regions inside $M_{i,\mathrm{ext}}$ containing $\partial M_{i,\mathrm{ext}}$, and let $\Sigma_i$ be the outer boundary of $\Omega_i$. Assume that $\operatorname{diam}(\Omega_i)\leq D$ and $|\Sigma_i|=A_0$. Lee–Sormani's conjecture. Then $\Omega_i\to\Omega_0$ in the intrinsic flat metric topology, where $\Omega_0$ is a ball in Euclidean space satisfying $|\partial\Omega_0|=A_0$. This is an intrinsic-flat formulation of stability for the positive mass theorem; the source presents it as a conjecture following the earlier Gromov–Hausdorff formulation.