Convergence of Manifolds and Metric Spaces with Boundary

TL;DR

This paper investigates the convergence behavior of sequences of manifolds and metric spaces with boundary, establishing conditions under which Gromov-Hausdorff and Sormani-Wenger Intrinsic Flat limits coincide, leading to rectifiable limit spaces.

Contribution

It introduces new conditions ensuring GH and SWIF limit spaces agree for manifolds with boundary, extending convergence results to broader classes of metric spaces.

Findings

GH and SWIF limits coincide under specified conditions

Limit spaces are countably rectifiable

Established GH compactness theorems for manifolds with boundary

Abstract

We study sequences of oriented Riemannian manifolds with boundary and, more generally, integral current spaces and metric spaces with boundary. {\color{blue}For a metric space, we define its boundary to be the completion of the space minus the space. We impose conditions on these spaces in order to get Gromov-Hausdorff (GH) subconvergence. By adding some other conditions} we prove theorems demonstrating that the Gromov-Hausdorff (GH) and Sormani-Wenger Intrinsic Flat (SWIF) limits of sequences of such metric spaces agree. Thus in particular the limit spaces we get are countably $\mathcal{H}^n$ rectifiable spaces. From these we derive GH compactness theorems for sequences of Riemannian manifolds with boundary where both the GH and SWIF limits agree. For sequences of Riemannian manifolds with boundary we only require nonnegative Ricci curvature, upper bounds on volume, noncollapsing conditions on the interior of the manifold and diameter controls on the level sets near the boundary.