Sharp Geometric and Functional Inequalities in Metric Measure Spaces with Lower Ricci Curvature Bounds

Abstract. For metric measure spaces verifying the reduced curvature-dimension condition CD∗(K,N) we prove a series of sharp functional inequalities under the additional assumption of essentially nonbranching. 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 RCD∗(K,N)-spaces, Finsler manifolds endowed with a strongly convex norm and satisfying lower Ricci curvature bounds, etc. In particular we prove Brunn-Minkowski inequality, p-spectral gap (or equivalently p-Poincaré inequality) for any p ∈ [1,∞), log-Sobolev inequality, Talagrand inequality and finally Sobolev inequality. All the results are proved in a sharp form involving an upper bound on the diameter of the space; if this extra sharpening is suppressed, all the previous inequalities for essentially non-branching CD∗(K,N) spaces take the same form of the corresponding ones holding for a weighted Riemannian manifold verifying curvature-dimension condition CD(K,N) in the sense of Bakry-Émery. In this sense inequalities are sharp. We also discuss the rigidity and almost rigidity statements associated to the p-spectral gap. Finally let us mention that for essentially non-branching metric measure spaces, the local curvaturedimension condition CDloc(K,N) is equivalent to the reduced curvature-dimension condition CD ∗(K,N). Therefore we also have shown that 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 CD(K,N).