Weak convergence of currents and cancellation

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 \({{\fancyscript H}^m}\)-rectifiability of the Gromov-Hausdorff limit of sequences of Lipschitz manifolds Mn all of which are λ-linearly locally contractible up to some scale r0. In the Appendix, we show that the Gromov-Hausdorff limit need not be countably \({{\fancyscript H}^m}\) -rectifiable if the Mn 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.