Inserted: 13 jan 2010
Last Updated: 3 feb 2010
Journal: Calc. Var. Partial Differential Equations
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.