by Luigi Ambrosio and Bernd Kirchheim

Acta Math. 185 (2000), no. 1, 1--80.

preprint, reprint, mathscinet review

Abstract: We develop a theory of currents in metric spaces which extends the classical theory of Federer--Fleming in euclidean spaces and in Riemannian manifolds. The main idea is to replace the duality with differential forms with the duality with (k + 1)-ples (f; 1 ; : : : ; k ) of Lipschitz functions, where k is the dimension of the current. We show, by a metric proof which is new even for currents in euclidean spaces, that the closure theorem and the boundary rectifiability theorem for integral currents hold in any complete metric space E. Moreover, we prove some existence results for a generalized Plateau problem in compact metric spaces and in some classes of Banach spaces, not necessarily finite dimensional.

There is also a Compactness Theorem and a Slicing Theorem in this paper. Note that this paper addresses weak convergence rather than flat convergence. Flat convergence is developed in "Flat Convergence for Integral Currents" by Stefan Wenger and is shown to be equivalent to weak convergence under some additional conditions. Key aspects of Ambrosio-Kirchheim's paper and prior work of Wenger are reviewed in the next paper:

by Christina Sormani and Stefan Wenger

(this preprint has been delayed as we update it and add results)

Abstract: Inspired by the Gromov-Hausdorff distance, we define the intrinsic flat distance
between oriented m dimensional Riemannian manifolds with boundary as:

d_F(M,N) = inf d^Z_F([f(M)], [h(N)] )
where f,h are isometric embeddings from M and N respectively into a complete metric space Z
and where [f(M)] and [h(N)] denote their images viewed as integral currents in Z
in the sense of Ambrosio-Kirchheim and the infimum is taken over all $f$, $h$ and $Z$.

We show the intrinsic flat distance between oriented Riemannian manifolds is zero iff
they have an orientation preserving isometry between them. Using the theory of
Ambrosio-Kirchheim, we study converging sequences of manifolds and their
limits, which are in a class of metric spaces that we call integral current spaces.
We describe the properties of such spaces including the fact that they are
countably H^m rectifiable spaces and give numerous examples.

"Compactness for manifolds and integral currents with bounded diameter and volume"

by Stefan Wenger

(arxiv preprint)

Abstract: By Gromov's compactness theorem for metric spaces, every uniformly compact sequence
of metric spaces admits an isometric embedding into a common compact metric space in
which a subsequence converges with respect to the Hausdorff distance. Working in the
class or oriented $k$-dimensional Riemannian manifolds and, more generally, integral
currents in metric spaces (in the sense of Ambrosio-Kirchheim) and replacing the
Hausdorff distance with the filling volume or flat distance, we prove an analogous
compactness theorem in which we replace equicompactness with uniform upper bounds on
volume and diameter for the sequence."

In particular, Wenger has shown that any sequence of oriented $k$ dimensional Riemannian
manifolds with a uniform upper bound on volume and on diameter has a subsequence which
converges in the intrinsic flat sense to an integral current space.
If the manifolds have boundary, he only needs to add an assumption
that the boundaries have a uniform upper bound on volume.

"Cancellation under weak convergence" (prior name: "Cancellation under flat convergence")

by Christina Sormani and Stefan Wenger

Appendix by Raanan Schul and Stefan Wenger

To appear in Calculus of Variations and Partial Differential Equations

(arxiv preprint) (reprint)