Volumes and limits of manifolds with Ricci curvature and mean curvature bounds

We consider smooth Riemannian manifolds with nonnegative Ricci curvature and smooth boundary. First we prove a global Laplacian comparison theorem in the barrier sense for the distance to the boundary. We apply this theorem to obtain volume estimates of the manifold and of regions of the manifold near the boundary depending upon an upper bound on the area and on the inward pointing mean curvature of the boundary. We prove that families of oriented manifolds with uniform bounds of this type are compact with respect to the Sormani–Wenger Intrinsic Flat (SWIF) distance.