Ricci curvature for metric-measure spaces via optimal transport
type: Item
instance of: scholarly article
title: Ricci curvature for metric-measure spaces via optimal transport
zbMATH Open document ID: 1178.53038
author: John Lott
author: Cédric Villani
published in: Annals of Mathematics. Second Series
publication date: 2009-07-14T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0412127
full work available at URL: p04.xhtml
review text: The main results of this very interesting paper, showing that there is a good notion of a compact measured length space having ``\(N\)-Ricci curvature bounded below by \(K\in \mathbb{R}\)'' for each \(N\in [1,+\infty]\) -- via optimal transport in the setting of length spaces -- which is accompanied with geometric and analytic consequences, have been presented in a previous paper of the first author [``Optimal transport and Ricci curvature for metric-measure spaces'', see our review of this paper, that appeared in Surveys in Differential Geometry 11, 229--257 (2007; Zbl 1155.53026)]. The authors provide here complete proofs of these results. In the case of Riemannian manifolds their definitions are equivalent to classical ones. There are many other aspects of the paper, and in particular, six appendices that contain technical and auxiliary results, explanations how to extend the results of the paper from the setting of compact measured length spaces to the setting of complete pointed locally compact measured length spaces, and some bibliographic notes on optimal transport and displacement convexity, which illustrate the authors theory in a convincing way.
Mathematics Subject Classification ID: 53C23
Mathematics Subject Classification ID: 53C21
Mathematics Subject Classification ID: 60B05
Mathematics Subject Classification ID: 60B10
Mathematics Subject Classification ID: 58C35
zbMATH DE Number: 5578758
zbMATH Keywords: length space
zbMATH Keywords: measured Gromov-Hausdorff convergence
zbMATH Keywords: Ricci curvature
zbMATH Keywords: optimal transport
zbMATH Keywords: Wasserstein space
zbMATH Keywords: displacement interpolation
zbMATH Keywords: displacement convexity
zbMATH Keywords: Wasserstein geodesic
zbMATH Keywords: Bishop-Gromov inequality
zbMATH Keywords: logarithmic Sobolev inequality
zbMATH Keywords: Talagrand inequality
zbMATH Keywords: Poincaré inequality
zbMATH Keywords: Riemannian manifold
zbMATH Keywords: Aleksandrov space
reviewed by: Mircea Craioveanu
MaRDI profile type: Publication
OpenAlex ID: W1968781546
Wikidata QID: Q56813466
arXiv ID: math/0412127
DOI: 10.4007/ANNALS.2009.169.903
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description: scientific article