Tetrahedra, Rectifiability and the Gromov-Hausdorff Convergence of Metric Spaces
Time and place
1 PM on Thursday, November 14th, 2013; NAC 6/113
Christina Sormani (Lehman College and CUNY Graduate Center)
Abstract
We will begin with a review of the Gromov-Hausdorff distance between metric spaces: a notion which is fundamental in both Geometric Analysis and Group Theory. Also fundamental to Geometric Analysis is the notion of Hausdorff measure and countably $H^m$ rectifiable metric spaces (a class of metric spaces upon which one can define differentiation almost everywhere). Even more smooth are the Riemannian manifolds where one can define second derivatives and study partial differential equations. Gromov's Compactness Theorem states that sequences of metric spaces with a uniform upper bound on diameter and on the numbers of disjoint balls contained in the space, have a converging subsequence. One can apply his theorem to study sequences of Riemannian manifolds; however, the limits may only be metric spaces, not Riemannian manifolds and not even countably $H^m$ rectifiable. Today we present a new theorem in which a sequence of Riemannian manifolds is assumed to have an upper bound on diameter and on volume and also a uniform condition on tetrahedra, and one finds a subsequence converges in the Gromov-Hausdorff sense to a countably $H^m$ rectifiable metric space. We do not present the proof but it involves tools from Geometric Measure Theory, work of Ambrosio-Kirchheim dating to 2000, as well as the notion of Intrinsic Flat convergence developed in joint work with Stefan Wenger in 2011.
