Sharp geometric and functional inequalities in metric measure spaces with lower Ricci curvature bounds

Abstract

For metric measure spaces satisfying the reduced curvature–dimension condition ${\mathsf{CD}}^{\ast }\left(K,N\right)$ we prove a series of sharp functional inequalities under the additional “essentially nonbranching” assumption. Examples of spaces entering this framework are (weighted) Riemannian manifolds satisfying lower Ricci curvature bounds and their measured Gromov Hausdorff limits, Alexandrov spaces satisfying lower curvature bounds and, more generally, ${\mathsf{RCD}}^{\ast }\left(K,N\right)$ spaces, Finsler manifolds endowed with a strongly convex norm and satisfying lower Ricci curvature bounds.

In particular we prove the Brunn–Minkowski inequality, the $p$–spectral gap (or equivalently the $p$–Poincaré inequality) for any $p\in \left[1,\infty \right)$, the log-Sobolev inequality, the Talagrand inequality and finally the Sobolev inequality.

All the results are proved in a sharp form involving an upper bound on the diameter of the space; all our inequalities for essentially nonbranching ${\mathsf{CD}}^{\ast }\left(K,N\right)$ spaces take the same form as the corresponding sharp ones known for a weighted Riemannian manifold satisfying the curvature–dimension condition $\mathsf{CD}\left(K,N\right)$ in the sense of Bakry and Émery. In this sense our inequalities are sharp. We also discuss the rigidity and almost rigidity statements associated to the $p$–spectral gap.

In particular, we have also shown that the sharp Brunn–Minkowski inequality in the global form can be deduced from the local curvature–dimension condition, providing a step towards (the long-standing problem of) globalization for the curvature–dimension condition $\mathsf{CD}\left(K,N\right)$.

To conclude, some of the results can be seen as answers to open problems proposed in Villani’s book Optimal transport.