Appears in collection : Summer School 2021 - Curvature Constraints and Spaces of Metrics

Given a closed and oriented manifold M and Riemannian tensors g0, g1, ... on M that satisfy g0 < gj, vol(M, gj)→vol (M, g0) and diam(M, gj)≤D we will see that (M, gj) converges to (M, g0) in the intrinsic flat sense. We also generalize this to the non-empty bundary setting. We remark that under the onditions we do not nexessarily obtain smooth, C0 or even Gromov-Hausdorff convergence. furthermore, these results can be applied to show stability of a class of tori and a class of complete and asymptotically flat manifolds. That is, any sequence of tori in the former class with almost nonnegative scalar curvature convergences to a flat tori, and any sequence of manifolds in the latter with ADM masses converging to zero converges to Euclidean space. [Based on joint work with Allen, Allen-Sormani and Cabrera Pacheco-Katterer].