Independence of synthetic curvature dimension conditions on transport distance exponent
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- by Afiny Akdemir, Andrew Colinet, Robert J. McCann, Fabio Cavalletti and Flavia Santarcangelo PDF
- Trans. Amer. Math. Soc. 374 (2021), 5877-5923 Request permission
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
- © Copyright 2021 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 374 (2021), 5877-5923
- MSC (2020): Primary 49Q22, 51Fxx
- DOI: https://doi.org/10.1090/tran/8413
- MathSciNet review: 4293791