Ricci curvature for metric-measure spaces via optimal transport
Annals of mathematics, Tome 169 (2009) no. 3, pp. 903-991
We define a notion of a measured length space $X$ having nonnegative $N$-Ricci curvature, for $N \in [1, \infty)$, or having $\infty$-Ricci curvature bounded below by $K$, for $K \in \mathbb{R}$. The definitions are in terms of the displacement convexity of certain functions on the associated Wasserstein metric space $P_2(X)$ of probability measures. We show that these properties are preserved under measured Gromov-Hausdorff limits. We give geometric and analytic consequences.
@article{10_4007_annals_2009_169_903,
author = {John Lott and Cedric Villani},
title = {Ricci curvature for metric-measure spaces via optimal transport},
journal = {Annals of mathematics},
pages = {903--991},
year = {2009},
volume = {169},
number = {3},
doi = {10.4007/annals.2009.169.903},
mrnumber = {2480619},
zbl = {1178.53038},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4007/annals.2009.169.903/}
}
TY - JOUR AU - John Lott AU - Cedric Villani TI - Ricci curvature for metric-measure spaces via optimal transport JO - Annals of mathematics PY - 2009 SP - 903 EP - 991 VL - 169 IS - 3 UR - http://geodesic.mathdoc.fr/articles/10.4007/annals.2009.169.903/ DO - 10.4007/annals.2009.169.903 LA - en ID - 10_4007_annals_2009_169_903 ER -
%0 Journal Article %A John Lott %A Cedric Villani %T Ricci curvature for metric-measure spaces via optimal transport %J Annals of mathematics %D 2009 %P 903-991 %V 169 %N 3 %U http://geodesic.mathdoc.fr/articles/10.4007/annals.2009.169.903/ %R 10.4007/annals.2009.169.903 %G en %F 10_4007_annals_2009_169_903
John Lott; Cedric Villani. Ricci curvature for metric-measure spaces via optimal transport. Annals of mathematics, Tome 169 (2009) no. 3, pp. 903-991. doi: 10.4007/annals.2009.169.903
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