Non-collapsed spaces with Ricci curvature bounded from below

Abstract

We propose a definition of non-collapsed space with Ricci curvature bounded from below and we prove the versions of Colding's volume convergence theorem and of Cheeger-Colding dimension gap estimate for RCD spaces.In particular this establishes the stability of non-collapsed spaces under non-collapsed Gromov-Hausdorff convergence. ContentsLott-Villani in [39] and Sturm in [44,45] introduced a synthetic notion of lower Ricci curvature bounds for metric measure spaces: their approach is based on suitable convexity properties for entropy-like functionals over the space of probability measures equipped with the quadratic Kantorovich distance W 2 . The classes of spaces that they introduced are called CD(K, N ), standing for lower Curvature bound by K ∈ R and upper Dimension bound by N ∈ [1, ∞] (in [39] only the cases K = 0 and N = ∞ have been considered). * SISSA, gdephili@sissa.it † SISSA, ngigli@sissa.itSince then the study of these classes of spaces has been a very flourishing research area, see for instance the surveys [48,46] and references therein. Among the various fine tunings of Lott-Sturm-Villani's proposal, we mention the definition of CD * (K, N ) spaces proposed by Bacher-Sturm in [10]: under minor technical conditions, this is locally equivalent to the CD(K, N ), has better local-to-global properties but a priori leads to slightly suboptimal constants in various geometric and functional inequalities (but see (1.2) below).Since the very beginning, one of the main research line has been, and still is, that of understanding the geometric properties of such spaces. Here fundamental ideas come from the theory of Ricci-limit spaces developed in the nineties by Cheeger and Colding [14,15,16,17,20]: one would like at least to replicate all their results in the synthetic framework, and then hopefully obtaining, thanks to the new point of view, new insights about both smooth and non-smooth objects having Ricci curvature bounded from below. In this direction it has been soon realized that the classes of CD/CD * (K, N ) spaces are not really suitable for the development of this program: the problem is that Finlser structures are included (see the last theorem in [47]) and for these Cheeger-Colding's results are not valid. For instance, the Cheeger-Colding-Gromoll splitting theorem fails in finite dimensional Banach spaces.Motivated by this problem the second author proposed in [27] to reinforce the Lott-Sturm-Villani condition with the functional-analytic notion of infinitesimal Hilbertianity: (X, d, m) is infinitesimal Hilbertian provided W 1,2 (X, d, m) is an Hilbert space.( 1.1) This definition is the result of a research program devoted to the understanding of the heat flow [24, 28, 3] on CD(K, N ) spaces, and in particular of the introduction of the class of RCD(K, ∞) spaces -R standing for Riemannian -in a collaboration with Ambrosio and Savaré [4]. In (1.1), W 1,2 (X, d, m) is the Sobolev space of real valued functions on X as introduced by Cheeger in [13] (see also the alternative, but equivalent, descripti...

Collapsed Ricci Limit Spaces as Non-Collapsed RCD Spaces

“…Assume that Conjecture 4.1 is true. If (a) holds, then it follows from a result of [12], which confirms a conjecture raised in [21], that (X, d, m) is an RCD(K, k) space, where k = dim(X, d, m). In particular, Conjecture 4.1 yields (b).…”

“…Let us mention that dim(X, d, m) is equal to N if (X, d, m) is a non-collapsed RCD(K, N ) space. It is conjectured in [21] that the converse implication is also true up to multiplication by a positive constant to the measure, that is: This conjecture is true if (X, d) is compact, which is proved in [34]. Note that since the RCD(K, N ) condition is unchanged under multiplication by a positive constant to the measure, if X, d, aH N is an RCD(K, N ) space, then X, d, H N is a non-collapsed RCD(K, N ) space.…”

“…On the other hand, a special class of RCD(K, N ) spaces, so-called non-collapsed RCD(K, N ) spaces, is proposed in [21] as the synthetic counterpart of non-collapsed Ricci limit spaces. An RCD(K, N ) space is said to be non-collapsed if m = H N holds.…”

Sphere theorems for RCD and stratified spaces

“…where H n is the n-dimensional Hausdorff measure. This is justified by using a recent result of the first author [H19] which confirms a conjecture by De Philippis-Gigli (see Remark 1.9 in [DePhG18]) in the compact setting. Then by combining this with a compactness result for non-collapsed RCD spaces by De Philippis-Gigli [DePhG18] and Ketterer's rigidity [K15a], we can show that our situation is reduced to the study of the following measured Gromov-Hausdorff convergent sequence of RCD(n − 1, n) spaces:…”

Weakly Noncollapsed RCD Spaces with Upper Curvature Bounds

“…it is equal to N . It was conjectured by Gigli and DePhilippis in [DPG18] that in this case up to a constant multiple m = H n i.e. up to rescaling the measure by a constant weakly non-collapsed spaces are non-collapsed.…”