Monotone Sequences of Metric Spaces with Compact Limits

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 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 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.

Keywords

Metric spaces / Riemannian / Gromov-Hausdorff / intrinsic flat

Acknowledgements

The authors would like to thank Wenchuan Tian and Changliang Wang for inspiring discussions of examples that motivated this paper. We would also like to thank the referee and students, Wai Ho Yeung and Kevin Bui, for careful reading of various parts of the first version of this paper and helpful feedback.

R. Perales’s research is partially funded by (Grant No. NSF DMS 1612049) on Geo-metric Compactness Theorems. C. Sormani’s research is partially funded by (Grant No. NSF DMS 1612049) on Geometric Compactness Theorems and a PSC-CUNY grant.

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