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

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Hölder Continuity and Injectivity of Optimal Maps (2011, background): “Under the weaker hypothesis (A3)w of Trudinger and Wang [51], which includes the cost c(x, y) = |x − y|2/2 (and whose necessity for regularity was shown by Loeper [42], see also [23]), such a result remained absent from the literature; the aim of this paper is to fill this gap, see Theorem 2.”
The Monge–Ampère equation and its link to optimal transportation (2013, unknown): “Indeed, as shown in [58], both the MTW condition and the convexity of the tangent cut-loci are necessary (and in some cases sufficient) conditions for the continuity of optimal transport maps on manifolds.”
Partial regularity for optimal transport maps (2012, methodology): “Indeed, as shown by Ma, Trudinger, and Wang [33], and Loeper [31], in addition to suitable convexity assumptions on the support of the target density (or on the cut locus of the manifold when supp(g) = M [24]), a very strong structural condition on the cost function, the so-called MTW condition, is ”