Weak convergence and cancellation

TL;DR

This paper investigates the relationship between weak limits of integral currents and Hausdorff limits of their supports in metric spaces, providing conditions to prevent cancellation and applying results to limits of Lipschitz manifolds.

Contribution

It establishes conditions ensuring the support of weak limits matches Hausdorff limits and applies these to prove rectifiability of Gromov-Hausdorff limits of certain manifolds.

Findings

Weak limits can be supported in a strict subset of the Hausdorff limit due to cancellation.

Sufficient topological conditions prevent cancellation, aligning weak and Hausdorff limits.

Gromov-Hausdorff limits of Lipschitz manifolds are countably rectifiable under linear local contractibility.

Abstract

In this article, we study the relationship between the weak limit of a sequence of integral currents in a metric space and the possible Hausdorff limit of the sequence of supports. Due to cancellation, the weak limit is in general supported in a strict subset of the Hausdorff limit. We exhibit sufficient conditions in terms of topology of the supports which ensure that no cancellation occurs and that the support of the weak limit agrees with the Hausdorff limit of the supports. We use our results to prove countable $\hm^m$-rectifiability of the Gromov-Hausdorff limit of sequences of Lipschitz manifolds $M_n$ all of which are $\lambda$-linearly locally contractible up to some scale $r_0$. In an appendix, written by Raanan Schul and the second author, it is shown that the Gromov-Hausdorff limit need not be countably $\hm^m$-rectifiable if the $M_n$ have a common local geometric contractibility function which is only concave (and not linear). We also relate our results to work of Cheeger-Colding on the limits of noncollapsing sequences of manifolds with nonnegative Ricci curvature.