Pseudo-Riemannian geometry calibrates optimal transportation

By Young-Heon Kim, Robert J. McCann, M. Warren. 2009

54 citations recorded by the source corpus.

Given a transportation cost c : M ◊ ¯ M ! R, optimal maps minimize the total cost of moving masses from M to ¯ M. We find, explicitly, a pseudo-metric and a calibration form on M ◊ ¯ 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.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/e94f62e6bdf4e105bfb54f591428f5b340e280c3

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In Applied Mathematics

In Mathematical Physics

Citations in the corpus

Hölder Continuity and Injectivity of Optimal Maps (2011, background): “5], and for further investigation of the pseudo-Riemannian aspects of optimal maps see [38] and [31]).”
Five lectures on optimal transportation: Geometry, regularity and applications (2010, unknown): “For almost every such y ∈ Sδ this provides the desired preimage x∗ ∈ G−1(y).”
Optimal transportation, topology and uniqueness (2010, background): “The first problem has become classical in the calculus of variations; it has deep connections to analysis [30,51,73,74,98], geometry [29,63,70,71,79,83,95,104], dynamics [7,10,59,82] and nonlinear partial differential equations [18,20,21,36,42, 72,102], as well as applications in physics [38,75,97],”