Tohoku Mathematical Journal
2011
December
SECOND SERIES VOL. 63, NO. 4

Tohoku Math. J.
63 (2011), 855-876

Title NECESSARY AND SUFFICIENT CONDITIONS FOR CONTINUITY OF OPTIMAL TRANSPORT MAPS ON RIEMANNIAN MANIFOLDS

Author Alessio Figalli, Ludovic Rifford and Cédric Villani

(Received October 6, 2010, revised June 13, 2011)
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.

2000 Mathematics Subject Classification. Primary 35J60; Secondary 53C21, 49Q20.
Key words and phrases. Nonlinear PDEs, optimal transport, riemannian geometry.

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