Rendiconti del Seminario Matematico della Università di Padova


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Volume 124, 2010, pp. 91–125
DOI: 10.4171/RSMUP/124-6

Metric Currents and Geometry of Wasserstein Spaces

Luca Granieri[1]

(1) Dipartimento di Matematica, Università degli Studi di Bari, Via E. Orabona 4, 70125, Bari, Italy

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 Pp(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 Pp(Tn) in its homology class.

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Granieri Luca: Metric Currents and Geometry of Wasserstein Spaces. Rend. Sem. Mat. Univ. Padova 124 (2010), 91-125. doi: 10.4171/RSMUP/124-6