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This paper is concerned with the study of the Monge optimal transport problemin sub-Riemannian manifolds where the cost is given by the square of thesub-Riemannian distance. Our aim is to extend previous results on existence anduniqueness of optimal transport maps to cases of sub-Riemannian structureswhich admit many singular minimizing geodesics. We treat here the case ofsub-Riemannian structures of rank two in dimension four.

Mass Transportation on sub-Riemannian structures of rank two in dimension four