Geometry, Topology and Dynamics Seminar
Stefan Wenger
UIC
Compactness for manifolds with bounded volume and diameter
Abstract: 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 filling volume or 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.
The new distance mentioned above was first introduced and studied by Christina Sormani and myself. In the talk, I will
explain the basic properties of this distance and will furthermore show how it can be used to prove countable k-rectifiability
for the Gromov-Hausdorff limit of sequences of certain Riemannian manifolds.
Monday January 12, 2009 at 3:00 PM in SEO 612