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Transport inequalities, gradient estimates, entropy and Ricci curvature
Author(s) -
von Renesse MaxK.,
Sturm KarlTheodor
Publication year - 2005
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.20060
Subject(s) - mathematics , ricci curvature , scalar curvature , semigroup , convexity , curvature , entropy (arrow of time) , riemannian manifold , mathematical analysis , brownian motion , manifold (fluid mechanics) , geometry , statistics , physics , mechanical engineering , quantum mechanics , financial economics , engineering , economics
We present various characterizations of uniform lower bounds for the Ricci curvature of a smooth Riemannian manifold M in terms of convexity properties of the entropy (considered as a function on the space of probability measures on M ) as well as in terms of transportation inequalities for volume measures, heat kernels, and Brownian motions and in terms of gradient estimates for the heat semigroup. © 2004 Wiley Periodicals, Inc.