The aim of this paper is to discuss convergence of pointed metric measure spaces in the absence of any compactness condition. We propose various definitions, and show that all of them are equivalent and that for doubling spaces these are also equivalent to the well-known measured Gromov-Hausdorff convergence.Then we show that the curvature conditions CD(K, ∞) and RCD(K, ∞) (Riemannian curvature dimension, RCD) are stable under this notion of convergence and that the heat flow passes to the limit as well, both in the Wasserstein and in the L 2 -framework. We also prove the variational convergence of Cheeger energies in the naturally adapted Γ-Mosco sense and the convergence of the spectra of the Laplacian in the case of spaces either uniformly bounded or satisfying the RCD(K, ∞) condition with K > 0. When applied to Riemannian manifolds, our results allow for sequences with diverging dimensions.

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

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Abstract. We prove that if (X, d, m) is a metric measure space with m(X) = 1 having (in a synthetic sense) Ricci curvature bounded from below by K > 0 and dimension bounded above by N ∈ [1, ∞), then the classic Lévy-Gromov isoperimetric inequality (together with the recent sharpening counterparts proved in the smooth setting by E. Milman for any K ∈ R, N ≥ 1 and upper diameter bounds) holds, i.e. the isoperimetric profile function of (X, d, m) is bounded from below by the isoperimetric profile of the model space. Moreover, if equality is attained for some volume v ∈ (0, 1) and K is strictly positive, then the space must be a spherical suspension and in this case we completely classify the isoperimetric regions. Finally we also establish the almost rigidity: if the equality is almost attained for some volume v ∈ (0, 1) and K is strictly positive, then the space must be mGH close to a spherical suspension. To our knowledge this is the first result about isoperimetric comparison for non smooth metric measure spaces satisfying Ricci curvature lower bounds. Examples of spaces fitting our assumptions include measured Gromov-Hausdorff limits of Riemannian manifolds satisfying Ricci curvature lower bounds, Alexandrov spaces with curvature bounded from below, Finsler manifolds endowed with a strongly convex norm and satisfying Ricci curvature lower bounds; the result seems new even in these celebrated classes of spaces.

Aim of this paper is to provide new characterizations of the curvature dimension condition in the context of metric measure spaces (X, d, m). On the geometric side, our new approach takes into account suitable weighted action functionals which provide the natural modulus of K-convexity when one investigates the convexity properties of N -dimensional entropies. On the side of diffusion semigroups and evolution variational inequalities, our new approach uses the nonlinear diffusion semigroup induced by the N -dimensional entropy, in place of the heat flow. Under suitable assumptions (most notably the quadraticity of Cheeger's energy relative to the metric measure structure) both approaches are shown to be equivalent to the strong CD * (K, N ) condition of Bacher-Sturm.

We prove that a metric measure space (X, d, m) satisfying finite dimensional lower Ricci curvature bounds and whose Sobolev space W 1,2 is Hilbert is rectifiable. That is, a RCD * (K, N)-space is rectifiable, and in particular for m-a.e. point the tangent cone is unique and euclidean of dimension at most N. The proof is based on a maximal function argument combined with an original Almost Splitting Theorem via estimates on the gradient of the excess. To this aim we also show a sharp integral Abresh-Gromoll type inequality on the excess function and an Abresh-Gromoll-type inequality on the gradient of the excess. The argument is new even in the smooth setting.1 [19,20,21,22] and more recently [27] by Colding and the second author. On the other hand, in the last ten years, there has been a surge of activity on general metric measure spaces (X, d, m) satisfying a lower Ricci curvature bound in some generalized sense. This investigation began with the seminal papers of Lott-Villani [40] and Sturm [46,47], though has been adapted considerably since the work of Bacher-Sturm [11] and Ambrosio-Gigli-Savaré [5,6]. The crucial property of any such definition is the compatibility with the smooth Riemannian case and the stability with respect to measured Gromov-Hausdorff convergence. While a great deal of progress has been made in this latter general framework, see for instance [3,5,6,7,9,10,11,15,17,29,30,31,32,34,35,36,42,43,44,48], the structure theory on such metric-measure spaces is still much less developed than in the case of smooth limits.The notion of lower Ricci curvature bound on a general metric-measure space comes with two subtleties. The first is that of dimension, and has been well understood since the work of Bakry-Emery [12]: in both the geometry and analysis of spaces with lower Ricci curvature bounds, it has become clear the correct statement is not that "X has Ricci curvature bounded from below by K", but that "X has N-dimensional Ricci curvature bounded from below by K". Such spaces are said to satisfy the (K, N)-Curvature Dimension condition, CD(K, N) for short; a variant of this is that of reduced curvature dimension bound, CD * (K, N). See [11,12,47] and Section 2 for more on this.The second subtle point, which is particularly relevant for this paper, is that the classical definition of a metric-measure space with lower Ricci curvature bounds allows for Finsler structures (see the last theorem in [48]), which after the aforementioned works of Cheeger-Colding are known not to appear as limits of smooth manifolds with Ricci curvature lower bounds. To address this issue, Ambrosio-Gigli-Savaré [6] introduced a more restrictive condition which rules out Finsler geometries while retaining the stability properties under measured Gromov-Hausdorff convergence, see also [3] for the present simplified axiomatization. In short, one studies the Sobolev space W 1,2 (X) of functions on X. This space is always a Banach space, and the imposed extra condition is that W 1,2 (X) is a Hilbert space. Equivalently, the Lapl...

We prove higher summability and regularity of Γ f for functions f in spaces satisfying the Bakry-Émery condition BE(K, ∞).As a byproduct, we obtain various equivalent weak formulations of BE(K, N ) and we prove the Local-to-Global property of the RCD * (K, N ) condition in locally compact metric measure spaces (X, d, m), without assuming a priori the non-branching condition on the metric space.

The goal of the paper is four-fold. In the setting of non-smooth spaces with Ricci curvature lower bounds (more precisely RCD(K, N ) metric measure spaces):• We establish a new principle relating lower Ricci curvature bounds to the preservation of Laplacian lower bounds under the evolution via the p-Hopf-Lax semigroup, for general exponents p ∈ [1, ∞).• We develop an instrinsic viscosity theory of Laplacian bounds.• We prove Laplacian bounds on the distance function from a set (locally) minimizing the perimeter; this corresponds to vanishing mean curvature in the smooth setting.• We initiate a regularity theory for boundaries of sets (locally) minimizing the perimeter. The class of RCD(K, N ) metric measure spaces includes as remarkable sub-classes: measured Gromov-Hausdorff limits of smooth manifolds with lower Ricci curvature bounds and finite dimensional Alexandrov spaces with lower sectional curvature bounds. Most of the results are new also in these frameworks.