"Convergence of Manifolds and Metric Spaces with Boundary"  

We study sequences of oriented Riemannian manifolds with boundary

and, more generally, integral current spaces and metric spaces 

with boundary. We prove theorems demonstrating when the Gromov-Hausdorff 

and Sormani-Wenger Intrinsic Flat limits of sequences of such

metric spaces agree.  Then for sequences of Riemannian manifolds with boundary we only require nonnegative Ricci curvature, upper bounds on volume, non collapsing conditions on the interior of the manifold and diameter controls on the level sets near the boundary to obtain converging subsequences where both limits coincide