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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Independence of synthetic curvature dimension conditions on transport distance exponent


Authors: Afiny Akdemir, Andrew Colinet, Robert J. McCann, Fabio Cavalletti and Flavia Santarcangelo
Journal: Trans. Amer. Math. Soc. 374 (2021), 5877-5923
MSC (2020): Primary 49Q22, 51Fxx
DOI: https://doi.org/10.1090/tran/8413
Published electronically: May 20, 2021
MathSciNet review: 4293791
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Abstract:

The celebrated Lott-Sturm-Villani theory of metric measure spaces furnishes synthetic notions of a Ricci curvature lower bound $K$ joint with an upper bound $N$ on the dimension. Their condition, called the Curvature-Dimension condition and denoted by $\mathsf {CD}(K,N)$, is formulated in terms of a modified displacement convexity of an entropy functional along $W_{2}$-Wasserstein geodesics. We show that the choice of the squared-distance function as transport cost does not influence the theory. By denoting with $\mathsf {CD}_{p}(K,N)$ the analogous condition but with the cost as the $p^{th}$ power of the distance, we show that $\mathsf {CD}_{p}(K,N)$ are all equivalent conditions for any $p>1$ — at least in spaces whose geodesics do not branch.

Following Cavalletti and Milman [The Globalization Theorem for the Curvature Dimension Condition, preprint, arXiv:1612.07623], we show that the trait d’union between all the seemingly unrelated $\mathsf {CD}_{p}(K,N)$ conditions is the needle decomposition or localization technique associated to the $L^{1}$-optimal transport problem. We also establish the local-to-global property of $\mathsf {CD}_{p}(K,N)$ spaces.


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Additional Information

Afiny Akdemir
Affiliation: Department of Mathematics, University of Toronto, Toronto, Ontario M5S 2E4, Canada
Email: afiny@math.toronto.edu

Andrew Colinet
Affiliation: Department of Mathematics, University of Toronto, Toronto, Ontario M5S 2E4, Canada
ORCID: 0000-0002-3948-3869
Email: andrew.colinet@mail.utoronto.ca

Robert J. McCann
Affiliation: Department of Mathematics, University of Toronto, Toronto, Ontario M5S 2E4, Canada
MR Author ID: 333976
ORCID: 0000-0003-3867-808X
Email: mccann@math.toronto.edu

Fabio Cavalletti
Affiliation: Mathematics Area, SISSA, Trieste, Italy
MR Author ID: 956139
Email: cavallet@sissa.it

Flavia Santarcangelo
Affiliation: Mathematics Area, SISSA, Trieste, Italy
MR Author ID: 1338762
Email: fsantarc@sissa.it

Received by editor(s): July 22, 2020
Received by editor(s) in revised form: January 12, 2021
Published electronically: May 20, 2021
Additional Notes: The third author’s research was supported in part by Natural Sciences and Engineering Research Council of Canada Discovery Grants RGPIN–2015–04383 and 2020–04162
Article copyright: © Copyright 2021 American Mathematical Society