Abstract: We prove generalized lower Ricci curvature bounds for warped products over complete Finsler manifolds. On the one hand our result covers a theorem of Bacher and Sturm concerning Euclidean and spherical cones (Bacher and Sturm [3]). On the other hand it can be seen in analogy to a result of Bishop and Alexander in the setting of Alexandrov spaces with curvature bounded from below (Alexander and Bishop, 2004 [2]). For the proof we combine techniques developed in these papers. Because the Finslerian warped product metric can degenerate we regard a warped product as metric measure space that is in general neither a Finsler manifold nor an Alexandrov space again but a space satisfying a curvature-dimension condition in the sense of Lott–Villani/Sturm.
Kazumasa Kuwada and Karl-Theodor Sturm Monotonicity of time-dependent transportation costs and coupling by reflection Potential Analysis, 39(3): 231-263 2013 http://dx.doi.org/10.1007/s11118-012-9327-4
Abstract: Based on a study of the coupling by reflection of diffusion processes, a new monotonicity in time of a time-dependent transportation cost between heat distribution is shown under Bakry-Émery’s curvature-dimension condition on a Riemannian manifold. The cost function comes from the total variation between heat distributions on spaceforms. As a corollary, we obtain a comparison theorem for the total variation between heat distributions. In addition, we show that our monotonicity is stable under the Gromov-Hausdorff convergence of the underlying space under a uniform curvature-dimension and diameter bound.