Metric Currents and Geometry of Wasserstein Spaces

Abstract

We investigate some geometric aspects of Wasserstein spaces through the continuity equation as worked out in mass transportation theory. By defining a suitable homology on the flat torus ${\mathbb{T}}^n$, we prove that the space ${\mathcal{P}}_p({\mathbb{T}}^n)$ has non-trivial homology in a metric sense. As a byproduct of the developed tools, we show that every parametrization of a Mather’s minimal measure on ${\mathbb{T}}^n$ corresponds to a mass minimizing metric current on ${\mathcal{P}}_p({\mathbb{T}}^n)$ in its homology class.