Seminar

A rectifiable metric space is a metric space (X,d) with a collection of bi-Lipschitz charts that cover all but a set of Hausdorff measure 0 of the space. Such a space can be endowed with an orientation and viewed as a rectifiable current space (X,d,T) where the T is called current structure and uses the charts to capture the notion of integration of forms (rigorously defined in work of Ambrosio-Kirchheim).  If the boundary of T, defined via Stoke's theorem, is also rectifiable, then (X,d,T) is called an integral current space.  This notion is defined in joint work with Stefan Wenger.

Riemannian manifolds of finite volume with cone singularities are examples of integral current spaces. If the manifold has a cusp singularity, the corresponding integral current space has the cusp removed. Here we will present the Tetrahedral Compactness Theorem which assumes certain uniform distance estimates on tetrahedra in a sequence of integral current spaces (or Riemannian manifolds) and a uniform upper bound on diameter and concludes that a subsequence converges in the Gromov-Hausdorff and Intrinsic Flat sense to an integral current space (in particular the limit is rectifiable and the same dimension as the sequence).