Gromov's compactness theorem for metric spaces asserts that every
uniformly compact sequence of metric spaces has a subsequence which
converges in the Gromov-Hausdorff sense to a compact metric space. I will
show in this talk that if one replaces the Hausdorff distance appearing in
Gromov's theorem by the flat distance then every sequence of oriented
k-dimensional Riemannian manifolds with a uniform bound on diameter and
volume has a subsequence which converges in this new distance to a
countably k-rectifiable metric space. I will then sketch some applications
of this theorem.
The new distance mentioned above was first introduced and studied by
Christina Sormani and myself. I will explain the basic properties of this
distance and its relationship with other distances.