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Sharp geometric and functional inequalities in metric measure spaces with lower Ricci curvature bounds
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Cavalletti, Fabio and Mondino, Andrea (2017) Sharp geometric and functional inequalities in metric measure spaces with lower Ricci curvature bounds. Geometry and Topology, 21 (1). pp. 603-645. doi:10.2140/gt.2017.21.603 ISSN 1465-3060.
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Official URL: https://doi.org/10.2140/gt.2017.21.603
Abstract
For metric measure spaces satisfying the reduced curvature–dimension condition CD*(K,N) 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, RCD*(K,N) 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), 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 non branching CD*(K,N) spaces take the same form as the corresponding sharp ones known for a weighted Riemannian manifold satisfying the curvature–dimension condition CD(K,N) 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 CD(K,N). To conclude, some of the results can be seen as answers to open problems proposed in Villani’s book Optimal transport.
Item Type: | Journal Article | ||||||
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Subjects: | Q Science > QA Mathematics | ||||||
Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||
Library of Congress Subject Headings (LCSH): | Spaces of measures, Ricci flow | ||||||
Journal or Publication Title: | Geometry and Topology | ||||||
Publisher: | Geometry & Topology Publications | ||||||
ISSN: | 1465-3060 | ||||||
Official Date: | 10 February 2017 | ||||||
Dates: |
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Volume: | 21 | ||||||
Number: | 1 | ||||||
Page Range: | pp. 603-645 | ||||||
DOI: | 10.2140/gt.2017.21.603 | ||||||
Status: | Peer Reviewed | ||||||
Publication Status: | Published | ||||||
Access rights to Published version: | Restricted or Subscription Access | ||||||
Date of first compliant deposit: | 26 October 2017 | ||||||
Date of first compliant Open Access: | 31 October 2017 | ||||||
RIOXX Funder/Project Grant: |
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