Abstract. Measure contraction property is one of the possible generalizations of Ricci curvature bound to more general metric measure spaces. In this paper, we discover sufficient conditions for a three dimensional contact subriemannian manifold to satisfy this property.

Abstract. We prove a Bishop volume comparison theorem and a Laplacian comparison theorem for three dimensional contact subriemannian manifolds with symmetry.

Measure contraction property is a synthetic Ricci curvature lower bound for metric measure spaces. We consider Sasakian manifolds with non-negative Tanaka-Webster Ricci curvature equipped with the metric measure space structure defined by the sub-Riemannian metric and the Popp measure. We show that these spaces satisfy the measure contraction property M CP (0, N ) for some positive integer N . We also show that the same result holds when the Sasakian manifold is equipped with a family of Riemannian metrics extending the sub-Riemannian one. Sub-Riemannian Manifolds and the Measure Contraction PropertyIn this section, we recall various notions and facts about contact sub-Riemannian manifolds and the measure contraction property.A sub-Riemannian manifold is a manifold M equipped with a distribution D and a positive definite metric ·, · SR defined on D. A smooth 8 , Y 2 )V = 0, (4) Rm(X 1 , V )X 2 = 0,

One of the most well-known results in the theory of optimal transportation is the equivalence between the convexity of the entropy functional with respect to the Riemannian Wasserstein metric and the Ricci curvature lower bound of the underlying Riemannian manifold. There are also generalizations of this result to the Finsler manifolds and manifolds with a Ricci flow background.In this paper, we study displacement interpolations from the point of view of Hamiltonian systems and give a unifying approach to the above mentioned results.

Abstract. The Ma-Trudinger-Wang curvature -or cross-curvature -is an object arising in the regularity theory of optimal transportation. If the transportation cost is derived from a Hamiltonian action, we show its cross-curvature can be expressed in terms of the associated Jacobi fields. Using this expression, we show the least action corresponding to a harmonic oscillator has zero crosscurvature, and in particular satisfies the necessary and sufficient condition (A3w) for the continuity of optimal maps. We go on to study gentle perturbations of the free action by a potential, and deduce conditions on the potential which guarantee either that the corresponding cost satisfies the more restrictive condition (A3s) of Ma, Trudinger and Wang, or in some cases has positive crosscurvature. In particular, the quartic potential of the anharmonic oscillator satisfies (A3s) in the perturbative regime.