Monotone Sequences of Metric Spaces with Compact Limits (Journal of Mathematical Study, 2025)
In this paper, we consider a fixed metric space (possibly an oriented Riemannian manifold with boundary) with an increasing sequence of distance functions anda uniform upper bound on diameter. When the fixed space endowed with the point-wise limit of these distances is compact, then there is uniform and Gromov-Hausdorff(GH) convergence to this space. When the fixed metric space also has an integral current structure of uniformly bounded total mass (as is true for an oriented Riemannianmanifold with boundary that has a uniform bound on total volume), we prove volumepreserving intrinsic flat convergence to a subset of the GH limit whose closure is thewhole GH limit. We provide a review of all notions and have a list of open questionsat the end.
Keywords: Metric spaces; Riemannian; Gromov-Hausdorff; intrinsic flat.