Intrinsic Flat Stability of Manifolds with Boundary where Volume Converges and Distance is Bounded Below

Given a compact, connected, and oriented manifold with boundary $M$ and a sequence of smooth Riemannian metrics defined on it, $g_j$, we prove volume preserving intrinsic flat convergence of the sequence to the smooth Riemannian metric $g_0$ provided $g_j$ always measures vectors strictly larger than or equal to $g_0$, the diameter of $g_j$ is uniformly bounded, the volume of $g_j$ converges to the volume of $g_0$, and $L^{\frac{m-1}{2}}$ convergence of the metrics restricted to the boundary. Many examples are reviewed which justify and explain the intuition behind these hypotheses. These examples also show that uniform, Lipschitz, and Gromov-Hausdorff convergence are not appropriate in this setting. Our results provide a new rigorous method of proving some special cases of the intrinsic flat stability of the positive mass theorem.