On the Ma–Trudinger–Wang curvature on surfaces

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The Monge–Ampère equation and its link to optimal transportation (2013, background): “ive" result in this case). Moreover only few manifolds are known to satisfy it (essentially just the ones listed in Section 4.3), and for instance it is known to fail on suciently at ellipsoids [57]. Let us also recall that, in the case of a Riemannian manifold, the MTW condition implies that t”
Partial regularity for optimal transport maps (2012, background): “Let us also mention that, up to now, the MTW condition is known to be satisfied only for very special classes of Riemannian manifolds, such as spheres, their products, their quotients and submersions, and their perturbations [32, 22, 10, 29, 25, 20, 11], and for instance it is known to fail on suffi”
Five lectures on optimal transportation: Geometry, regularity and applications (2010, background): “Thus −(vc, v) is a better candidate for a maximizer than −(u, v).”
NECESSARY AND SUFFICIENT CONDITIONS FOR CONTINUITY OF OPTIMAL TRANSPORT MAPS ON RIEMANNIAN MANIFOLDS (2011, methodology): “Now as far as the proofs are concerned, many ingredients are recycled from previous works [14, 15, 16, 22, 27], however they are improved and combined with new technical results, such as Lemma B.”