Metric measure spaces with Riemannian Ricci curvature bounded from below

Abstract

In this paper we introduce a synthetic notion of Riemannian Ricci bounds from below for metric measure spaces (X, d, m) which is stable under measured Gromov-Hausdorff convergence and rules out Finsler geometries. It can be given in terms of an enforcement of the Lott, Sturm and Villani geodesic convexity condition for the entropy coupled with the linearity of the heat flow. Besides stability, it enjoys the same tensorization, globalto-local and local-to-global properties. In these spaces, that we call RCD(K, ∞) spaces, we prove that the heat flow (which can be equivalently characterized either as the flow associated to the Dirichlet form, or as the Wasserstein gradient flow of the entropy) satisfies Wasserstein contraction estimates and several regularity properties, in particular Bakry-Emery estimates and the L ∞ − Lip Feller regularization. We also prove that the distance induced by the Dirichlet form coincides with d, that the local energy measure has density given by the square of Cheeger's relaxed slope and, as a consequence, that the underlying Brownian motion has continuous paths. All these results are obtained independently of Poincaré and doubling assumptions on the metric measure structure and therefore apply also to spaces which are not locally compact, as the infinite-dimensional ones.

Spectral convergence under bounded Ricci curvature

“…We generalize this observation to our nonsmooth setting based on the regularity theory of the heat flow given in [AGS14] by Ambrosio-Gigli-Savaré. A key idea is to use test functions instead of smooth functions.…”

“…The following is a direct consequence of the regularity theory of the heat flow from [AGS14] (thus, it holds on an RCD-space).…”

“…Claim 3.2 follows directly from the regularity theory of the heat flow in [AGS14]. For reader's convenience, we give a proof in the case when r = 1, s = 0.…”

“…See for instance [AGS14,LV09,St06a,St06b] for pionear works by Ambrosio-Gigli-Savaré, Lott-Villani, and Sturm. Roughly speaking, a metric measure space (Y, υ) is said to be an CD(K, N)-space if the Ricci curvature is bounded below by K and the dimension is bounded above by N, where K ∈ R and N ∈ (0, ∞].…”

The Isometry Group of an RCD ∗ Space is Lie

“…Guijarro and Santos-Rodríguez adapted the approach of [12] in which the existence of a splitting theorem is indispensable, thus only RCD * -spaces are contemplated. In contrast, besides considering independently the groups ISO(X) and ISO m (X), 2 we present a more general and direct method, which allows us to achieve the characterization stated in Theorem 1.1. Consequently, the major practical advantage is that our results are valid in a larger class of metric measure spaces: we are not restricted to work with spaces having solely Eulcidean tangents or in which a splitting theorem holds, see Remark 4.6.…”