In this talk we will deal with intrinsic flat convergence defined by Sormani and Wenger using work of Ambrosio and Kirchheim. We will show that given a closed and oriented manifold $M$ and Riemannian tensors $g_0 \leq g_j$ on $M$ that satisfy $vol(M, g_j)\to vol(M,g_0)$ and $diam(M,g_j)\leq D$ then $(M,g_j)$ converges to $(M,g_0)$ in the intrinsic flat sense. We note that under these conditions we do not necessarily obtain Gromov-Hausdorff convergence. We will show an analogous convergence result for manifolds with boundary. These results can be applied to show the stability of a class of tori with almost nonnegative scalar curvature and the stability of the positive mass theorem for a particular class of manifolds.

Based on joint work with Allen-Sormani and Allen.