Japanese Journal of Mathematics

, Volume 4, Issue 1, pp 27–45 | Cite as

Ricci curvature and measures



In the last thirty years three a priori very different fields of mathematics, optimal transport theory, Riemannian geometry and probability theory, have come together in a remarkable way, leading to a very substantial improvement of our understanding of what may look like a very specific question, namely the analysis of spaces whose Ricci curvature admits a lower bound. The purpose of these lectures is, starting from the classical context, to present the basics of the three fields that lead to an interesting generalisation of the concepts, and to highlight some of the most striking new developments.

Keywords and phrases:

Ricci curvature geometry of spaces of measures Bakry–Émery estimate optimal transport theory lower bounds on curvature Wasserstein distances metric measured spaces Gromov–Hausdorff topology entropy functionals 

Mathematics Subject Classification (2000):

58J65 53C21 49J20 28D05 90B06 60E15 


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Copyright information

© The Mathematical Society of Japan and Springer Japan 2009

Authors and Affiliations

  1. 1.Centre national de la recherche scientifique–Institut des Hautes Études ScientifiquesBures-sur-YvetteFrance

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