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Given a compact, connected, and oriented manifold with boundary $M$ and asequence of smooth Riemannian metrics defined on it, $g_j$, we prove volumepreserving intrinsic flat convergence of the sequence to the smooth Riemannianmetric $g_0$ provided $g_j$ always measures vectors strictly larger than orequal 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 themetrics restricted to the boundary. Many examples are reviewed which justifyand explain the intuition behind these hypotheses. These examples also showthat uniform, Lipschitz, and Gromov-Hausdorff convergence are not appropriatein this setting. Our results provide a new rigorous method of proving somespecial cases of the intrinsic flat stability of the positive mass theorem.

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