Optimal Transportation and Ricci Curvature for Metric Measure Spaces

We introduce and analyze generalized Ricci curvature bounds for metric measure spaces $(M,d,m)$, based on convexity properties of the relative entropy $Ent(. | m)$. For Riemannian manifolds, $Curv(M,d,m)\ge K$ if and only if $Ric_M\ge K $ on $M$. For the Wiener space, $Curv(M,d,m)=1$.

One of the main results is that these lower curvature bounds are stable under (e.g. measured Gromov-Hausdorff) convergence.

Moreover, we introduce a curvature-dimension condition CD$(K,N)$ being more restrictive than the curvature bound $Curv(M,d,m)\ge K$. For Riemannian manifolds, CD$(K,N)$ is {equivalent} to $\mbox{\rm Ric}_M(\xi,\xi)\ge K\cdot |\xi|^2$ and $\mbox{\rm dim}(M)\le N$.

Condition CD$(K,N)$ implies sharp version of the Brunn-Minkowski inequality, of the Bishop-Gromov volume comparison theorem and of the Bonnet-Myers theorem. Moreover, it allows to construct canonical Dirichlet forms with {Gaussian upper and lower bounds} for the corresponding heat kernels.