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 $C{D}^{\ast }(K,N)$ we prove a series of sharp functional inequalities under the additional assumption of essentially non-branching. 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 $RC{D}^{\ast }(K,N)$-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 $p\in [1,\infty )$, 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 $C{D}^{\ast }(K,N)$ 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 curvature-dimension condition $C{D}_{loc}(K,N)$ is equivalent to the reduced curvature-dimension condition $C{D}^{\ast }(K,N)$. Therefore we also have shown that sharp Brunn-Minkowski inequality in the \emph{global} form can be deduced from the \emph{local} curvature-dimension condition, providing a step towards (the long-standing problem of) globalization for the curvature-dimension condition $CD(K,N)$.

accepted for publication "Geom. Topol." arXiv admin note: text overlap with arXiv:1502.06465

References in corpus (2)

• Sharp and rigid isoperimetric inequalities in metric-measure spaces with lower Ricci curvature bounds
• Optimal maps in essentially non-branching spaces

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