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 T n , we prove that the space p (T n ) has nontrivial homology in a metric sense. As a byproduct of the developed tools, we show that every parametrization of a Mather's minimal measure on T n corresponds to a mass minimizing metric current on p (T n ) in its homology class.