On the Ma-Trudinger-Wang curvature on surfaces

Figalli, Alessio; Rifford, Ludovic; Villani, Cédric
November 2010
Calculus of Variations & Partial Differential Equations;Nov2010, Vol. 39 Issue 3/4, p307
Academic Journal
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 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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