Riemannian Ricci curvature lower bounds in metric measure spaces with 𝜎-finite measure

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, 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 Lott-Villani, in the case N = ∞, to the Bakry-Emery 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 above-mentioned 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 σ-finite 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. enforced by adding the assumption that the so-called Cheeger energy (playing here the role of the classical Dirichlet energy) is quadratic.More precisely, the class of RCD(K, ∞) spaces of [4] can be defined in three equivalent ways thanks to this equivalence result (see §2.3 for the precise formulation of gradient flows involved here, in the metric sense and in the EV I K sense):Theorem 1.1 ([4]). Let (X, d, m) be a metric measure space with (X, d) complete and separable, m(X) ∈ (0, ∞) and supp m = X. Then the following are equivalent:m) is a strict CD(K, ∞) space and Ch is a quadratic form on L 2 (X, m). (iii) (X, d, m) is a length space and any μ ∈ P 2 (X) is the starting point of an EV I K gradient flow of Ent m . This equivalence is crucial for the study of the spaces RCD(K, ∞): for instance the fine properties of the heat flow and the Bakry-Emery condition obtained in [4] need (ii), while the stability of RCD(K, ∞) spaces under Sturm's convergence [33] of metric measure spaces (a variant of measured Gromov-Hausdorff convergence) depends in a crucial way on (iii) and on the stability properties of EV

The Isometry Group of an RCD ∗ Space is Lie

“…For example, Myers and Steenrod proved this fact for Riemannian Manifolds in [26], Fukaya and Yamaguchi for Alexandrov spaces with curvature bounded by above and by Gerardo Sosa gsosa@mis.mpg.de 1 Max Planck Institute for Mathematics in the Sciences, Leipzig 04103, Germany below in [12,36], and Cheeger, Colding, and Naber in the case of Ricci Limit spaces in [4,6]. In contrast, there exist metric spaces for which ISO(X) is not a Lie group, see for instance Examples 5.1 and 5.2.…”

“…A remarkable result of Gleason and Yamabe in the early 1950's asserts that a locally compact, topological group is not a Lie group if and only if every neighborhood of the identity has a non-trivial subgroup. 1 If a group has this property we say that it has the small subgroup property, ssp, for short. The strategy is to show the contrapositive statement in Theorem 1.1.…”

“…Alternatively, we present the results that show that these spaces satisfy the assumptions of Theorem 1.4. Nevertheless, let's do recall that the RCD * -condition condition was introduced in [2] and further developed in [1] and [9] to which the reader is referred to for a comprehensive discussion on the subject. The RCD * -condition itself couples a curvature-dimension condition with an infinitesimal Riemannian behavior known as infinitesimally Hilbertianity.…”

“…The Isometry Group of an RCD * -Space is Liê f ( 1 ) have distinct starting points than geodesics contained in 2 and moreover, different terminal points than geodesics in 1 . We claim that π ∈ OptGeo(μ 0 , μ 1 ) is also a dynamical plan.…”

Bakry–Émery curvature-dimension condition and Riemannian Ricci curvature bounds

“…One of the most important results of [5] (see also [2] for general σ-finite measures) is that CD(K, ∞) spaces with a quadratic Cheeger energy can be equivalently characterized as those metric measure spaces where there exists the Wasserstein gradient flow (H t ) t≥0 of the entropy functional (1.5) in the EVI K -sense. This condition means that for all initial data µ ∈ P 2 (X) with supp µ ⊂ supp m there exists a locally Lipschitz curve t → H t µ ∈ P 2 (X) satisfying the evolution variational inequality: for a.e.…”