Intrinsic flat Arzela–Ascoli theorems

One of the most powerful theorems in metric geometry is the Arzela–Ascoli Theorem which provides a continuous limit for sequences of equicontinuous functions between two compact spaces. This theorem has been extended by Gromov and Grove–Petersen to sequences of functions with varying domains and ranges where the domains and the ranges respectively converge in the Gromov–Hausdorff sense to compact limit spaces. However such a powerful theorem does not hold when the domains and ranges only converge in the intrinsic flat sense due to the possible disappearance of points in the limit.

In this paper two Arzela–Ascoli Theorems are proven for intrinsic flat converging sequences of manifolds: one for uniformly Lipschitz functions with fixed range whose domains are converging in the intrinsic flat sense, and one for sequences of uniformly local isometries between spaces which are converging in the intrinsic flat sense. A basic Bolzano–Weierstrass Theorem is proven for sequences of points in such sequences of spaces. In addition it is proven that when a sequence of manifolds has a precompact intrinsic flat limit then the metric completion of the limit is the Gromov–Hausdorff limit of regions within those manifolds. Applications and suggested applications of these results are described in the final section of this paper.

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This material is based upon work supported by the National Science Foundation under Grant No. 0932078 000, while the author was serving as a Visiting Research Professor at the Mathematical Sciences Research Institute in Berkeley, California, during the Fall of 2013. The author’s research was also funded by an individual research grant NSF-DMS-1309360 and a PSC-CUNY Research Grant.

Received 24 April 2015

Published 29 March 2019