Riemannian Ricci curvature lower bounds in metric measure spaces with σfinite measure
Ambrosio, L., Gigli, N., Mondino, A., & Rajala, T. (2015). Riemannian Ricci curvature lower bounds in metric measure spaces with σfinite measure. Transactions of the American Mathematical Society, 367(7), 46614701. https://doi.org/10.1090/S00029947201506111X
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© 2015 American Mathematical Society. First published in Transactions of the American Mathematical Society in March 4, 2015, published by the American Mathematical Society. Published in this repository with the kind permission of the publisher.
In a prior work of the first two authors with Savar´e, a new Riemannian
notion of a lower bound for Ricci curvature in the class of metric measure
spaces (X, d, m) was introduced, and the corresponding class of spaces was
denoted by RCD(K,∞). This notion relates the CD(K, N) theory of Sturm
and LottVillani, in the case N = ∞, to the BakryEmery approach. In this
prior work the RCD(K,∞) property is defined in three equivalent ways and
several properties of RCD(K,∞) 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 abovementioned work
only finite reference measures m have been considered. The goal of this paper
is twofold: on one side we extend these results to general σfinite spaces, and
on the other we remove a technical assumption that appeared in the earlier
work concerning a strengthening of the CD(K,∞) 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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