Necessary and sufficient conditions for continuity of optimal transport maps on Riemannian manifolds

Abstract : In this paper we continue the investigation of the regularity of optimal transport maps on Riemannian manifolds, in relation with the geometric conditions of Ma-Trudinger-Wang and the geometry of the cut locus. We derive some sufficient and some necessary conditions to ensure that the optimal transport map is always continuous. In dimension two, we can sharpen our result into a necessary and sufficient condition. We also provide some sufficient conditions for regularity, and review existing results.
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Submitted on : Monday, April 7, 2014 - 2:18:01 PM
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Alessio Figalli, Ludovic Rifford, Cédric Villani. Necessary and sufficient conditions for continuity of optimal transport maps on Riemannian manifolds. Tohoku mathematical journal, Mathematical Institute of Tohoku University, 2011, 63 (4), pp.855-876. ⟨10.2748/tmj/1325886291⟩. ⟨hal-00974770⟩

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