Monotone Sequences of Metric Spaces with Compact Limits

TL;DR

This paper investigates the convergence properties of monotone sequences of metric spaces with compact limits, establishing conditions for Gromov-Hausdorff and intrinsic flat convergence, especially in the context of Riemannian manifolds.

Contribution

It extends the understanding of convergence of metric spaces with increasing distances and bounded diameter, including volume-preserving convergence in the integral current setting.

Findings

Gromov-Hausdorff convergence under compactness assumptions

Intrinsic flat convergence with volume preservation

Conditions for convergence of monotone metric space sequences

Abstract

In this paper, we consider a fixed metric space (possibly an oriented Riemannian manifold with boundary) with an increasing sequence of distance functions and a uniform upper bound on diameter. When the metric space endowed with the pointwise limit of these distances is compact, then there is uniform and Gromov-Hausdorff (GH) convergence to this limit. When the metric space also has an integral current structure of uniformly bounded total mass (as is true for an oriented Riemannian manifold with boundary that has a uniform bound on total volume), we prove volume preserving intrinsic flat convergence to a subset of the GH limit whose closure is the whole GH limit. We provide a review of all notions and have a list of open questions at the end. Dedicated to Xiaochun Rong.