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 Tn, we prove that the space _P_p(Tn) 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 Tn corresponds to a mass minimizing metric current on _P_p(Tn) in its homology class.
Cite this article
Luca Granieri, Metric Currents and Geometry of Wasserstein Spaces. Rend. Sem. Mat. Univ. Padova 124 (2010), pp. 91–125