Seminar

In this talk we will study sequences of Riemannian manifolds with boundary and sequences of Alexandrov spaces.  We will prove a Gromov-Hausdorff precompactness theorem for manifolds with boundary. Then we demonstrate when the Gromov-Hausdorff and the Intrinsic Flat limits of sequences of either Riemannian manifolds with boundary or Alexandrov spaces agree. 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. In the Alexandrov case we assume non negative curvature, upper bounds on the diameter, and that the spaces are endowed with an integral current structure with weight function equal to 1.