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On the Ma-Trudinger-Wang condition on surfaces

Abstract : We investigate the properties of the Ma-Trudinger-Wang nonlocal curvature tensor in the case of surfaces. In particular, we prove that a strict form of the Ma-Trudinger-Wang condition is stable under C 4 perturbation if the nonfocal domains are uniformly convex; and we present new examples of positively curved surfaces which do not satisfy the Ma-Trudinger-Wang condition. As a corollary of our results, optimal transport maps on a "sufficiently flat" ellipsoid are in general nonsmooth.
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Alessio Figalli, Ludovic Rifford, Cédric Villani. On the Ma-Trudinger-Wang condition on surfaces. Calculus of Variations and Partial Differential Equations, Springer Verlag, 2010, 39 (3-4), pp.307. ⟨hal-00923304⟩

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