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# Mathematical Research Letters

## Volume 17 (2010)

### Number 6

### Pseudo-Riemannian geometry calibrates optimal transportation

Pages: 1183 – 1197

DOI: http://dx.doi.org/10.4310/MRL.2010.v17.n6.a16

#### Authors

#### Abstract

Given a transportation cost $c:M\times\bar{M}\rightarrow\mathbf{R}$, optimal maps minimize the total cost of moving masses from $M$ to $\bar{M}$. We find, explicitly, a pseudo-metric and a calibration form on $M\times\bar{M}$ such that the graph of an optimal map is a calibrated maximal submanifold, and hence has zero mean curvature. We define the mass of space-like currents in spaces with indefinite metrics.