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Riemannian Ricci curvature lower bounds in metric measure spaces with $ \sigma$-finite measure


Authors: Luigi Ambrosio, Nicola Gigli, Andrea Mondino and Tapio Rajala
Journal: Trans. Amer. Math. Soc. 367 (2015), 4661-4701
MSC (2010): Primary 49J52, 49Q20, 58J35, 31C25, 35K90
DOI: https://doi.org/10.1090/S0002-9947-2015-06111-X
Published electronically: March 4, 2015
MathSciNet review: 3335397
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Abstract: In a prior work of the first two authors with Savaré, a new Riemannian notion of a lower bound for Ricci curvature in the class of metric measure spaces $ (X,\textsf {d},\mathbf {m})$ was introduced, and the corresponding class of spaces was denoted by $ RCD(K,\infty )$. This notion relates the $ CD(K,N)$ theory of Sturm and Lott-Villani, in the case $ N=\infty $, to the Bakry-Emery approach. In this prior work the $ RCD(K,\infty )$ property is defined in three equivalent ways and several properties of $ RCD(K,\infty )$ spaces, including the regularization properties of the heat flow, the connections with the theory of Dirichlet forms and the stability under tensor products, are provided. In the above-mentioned work only finite reference measures $ \mathbf {m}$ have been considered. The goal of this paper is twofold: on one side we extend these results to general $ \sigma $-finite spaces, and on the other we remove a technical assumption that appeared in the earlier work concerning a strengthening of the $ CD(K,\infty )$ condition. This more general class of spaces includes Euclidean spaces endowed with Lebesgue measure, complete noncompact Riemannian manifolds with bounded geometry and the pointed metric measure limits of manifolds with lower Ricci curvature bounds.


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Additional Information

Luigi Ambrosio
Affiliation: Dipartimento di Matematica, Scuola Normale Superiore, Pisa, Italy
Email: luigi.ambrosio@sns.it

Nicola Gigli
Affiliation: Département de Mathématiques, University of Nice, Nice, France
Email: ngigli@sissa.it

Andrea Mondino
Affiliation: Dipartimento di Matematica, Scuola Normale Superiore, Pisa, Italy
Email: andrea.mondino@math.ethz.ch

Tapio Rajala
Affiliation: Department of Mathematics, University of Jyväskylä, FI-40014 Jyväskylä, Finland
Email: tapio.m.rajala@jyu.fi

DOI: https://doi.org/10.1090/S0002-9947-2015-06111-X
Received by editor(s): August 27, 2012
Received by editor(s) in revised form: February 13, 2013
Published electronically: March 4, 2015
Article copyright: © Copyright 2015 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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