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Pseudo-Riemann geometry calibrates optimal transportation. (English) Zbl 1222.49059

Summary: Given a transportation cost \(c:M\times\overline{M}\rightarrow\mathbb R\), optimal maps minimize the total cost of moving masses from \(M\) to \(\overline{M}\). We find, explicitly, a pseudo-metric and a calibration form on \(M\times\overline{M}\) 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.

MSC:

49Q20 Variational problems in a geometric measure-theoretic setting
53B21 Methods of local Riemannian geometry
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