Huisken–Ilmanen stability conjecture for the positive mass theorem

Let $(M_i,g_i)$ be a sequence of asymptotically flat $3$-manifolds with nonnegative scalar curvature, and let $m(g_i)$ denote their ADM masses, with $m(g_i)\to 0$. Huisken–Ilmanen stability conjecture. There are sets $Z_i\subset M_i$ such that

$$\operatorname{Area}(\partial Z_i)\to 0$$

and $(M_i\setminus Z_i,g_i)$ converges to Euclidean $3$-space in the Gromov–Hausdorff topology. This is the earliest conjecture concerning stability of the three-dimensional Riemannian Positive Mass Theorem, which asserts rigidity at zero mass. The statement allows small exceptional regions $Z_i$, reflecting the fact that small mass alone does not imply Gromov–Hausdorff closeness to Euclidean space.