New stability results for sequences of metric measure spaces with uniform Ricci bounds from below

Abstract

The aim of this paper is to provide new stability results for sequences of metric measure spaces (X i , d i , m i ) convergent in the measured Gromov-Hausdorff sense. By adopting the so-called extrinsic approach of embedding all metric spaces into a common one (X, d), we extend the results of [GMS13] by providing Mosco convergence of Cheeger's energies and compactness theorems in the whole range of Sobolev spaces H 1,p , including the space BV , and even with a variable exponent p i ∈ [1, ∞]. In addition, building on [AST16], we provide local convergence results for gradient derivations. We use these tools to improve the spectral stability results, previously known for p > 1 and for Ricci limit spaces, getting continuity of Cheeger's constant. In the dimensional case N < ∞, we improve some rigidity and almost rigidity results in [K15a, K15b, CM15a, CM15b]. On the basis of the second-order calculus in [G15b], in the class of RCD(K, ∞) spaces we provide stability results for Hessians and W 2,2 functions and we treat the stability of the Bakry-Émery condition BE(K, N ) and of Ric ≥ KI, with K and N not necessarily constant. * Scuola Normale Superiore, 11 Dimensional stability results 38 that in the class of smooth weighted n-dimensional Riemannian manifolds (M n , d, e −V vol M n ) the RCD * (K, N ) condition, n ≤ N , is equivalent toBy combining the continuity of (1.1) with the compactness property of the class of RCD * (K, N )-spaces w.r.t. the mGH convergence, we also establish a uniform boundwhere C i are positive constants depending only on K, N < ∞, and two-sided bounds of the diameter, i.e. C i do not depend on p (Proposition 11.1). Suspension theorems. The second application is related to almost spherical suspension theorems of positive Ricci curvature. For simplicity we discuss here only the case when N ≥ 2 is an integer, but our results (as those in [St06], [K15a], [K15b], [CM15b]) cover also the case N ∈ (1, ∞). In [CM15b] Cavalletti-Mondino proved that for any RCD * (N −1, N )space, the quantity (1.1) is greater than or equal than λ 1,p (S N , d, m N ) 1/p for any p ∈ [1, ∞), where S N is the unit sphere in R N +1 , d is the standard metric of the sectional curvature 1, and m N is the N -dimensional Hausdorff measure. Moreover, equality implies that the metric measure space is isomorphic to a spherical suspension. Under our notation (1.2) as above, this observation is also true when p = ∞, which corresponds to the Bonnet-Myers theorem in our setting (see [St06] by Sturm). Note that [CM15b] also provides rigidity results as the following one: for a fixed p ∈ [1, ∞], if (λ 1,p ) 1/p is close to λ 1,p (S N , d, m N ) 1/p , then the space is Gromov-Hausdorff close to the spherical suspension of a compact metric space, a so-called almost spherical suspension theorem. The converse is known for p ∈ {2, ∞} in [K15a, K15b] by Ketterer and we extend the result to general p; in addition, combining this with the joint spectral continuity result we can remove the p-dependence in the almost spherical suspension theore...

Collapsed Ricci Limit Spaces as Non-Collapsed RCD Spaces

“…This is to apply a geometric flow defined by embedding maps in L 2 via the global heat kernel. 5 Such embedding maps are introduced and studied first in [11] for closed Riemannian manifolds. Recently in [7] this observation is generalized to RCD(K, N ) spaces by using stability results of Sobolev functions with repect to the pmGH convergence proved in [5,6].…”

“…5 Such embedding maps are introduced and studied first in [11] for closed Riemannian manifolds. Recently in [7] this observation is generalized to RCD(K, N ) spaces by using stability results of Sobolev functions with repect to the pmGH convergence proved in [5,6]. It seems to the author that this technique, using a geometric flow, is useful for all conjectures proposed in this paper even in the case when (X, d) is non-compact.…”

Gaussian-type isoperimetric inequalities in RCD $(K, \infty)$ probability spaces for positive $K$

“…In addition, (2.23) and a standard chain rule provide the existence for L 1 -a.e. t 0 ∈ (0, T ) of the strong L 1 (X, m) derivative of G, given by 6) with g = H T −t 0 f and Ψ = Ψ(t 0 , g, Γg). Notice also that, by gradient contractivity,…”

“…One of the motivations for writing this note comes from [6], where the isoperimetric result of this paper is used to prove new compactness results in variable Sobolev spaces Let us mention that the isoperimetric inequality for Gaussian measures is originally due to C. Borell [14] and V. Sudakov -B. Cirel'son [25]. The functional form of the Gaussian isoperimetric inequality presented in Theorem 4.1 was introduced by S. Bobkov [13] who established it first on the two-point space and then in the limit for the Gaussian measure by the central limit Theorem.…”

Angles between Curves in Metric Measure Spaces

“…x the tangent cone is unique and euclidean [21,27]. From the technical point of view we also make use of the ne convergence results for Sobolev functions proved in [4,20], and we prove estimates on harmonic approximations of distance functions (see in particular Proposition 4.3). Harmonic approximations of distance functions are well known for smooth Riemannian manifolds with lower Ricci curvature bounds, and are indeed one of the key technical tools in the Cheeger-Colding theory of Ricci limit spaces [12][13][14]; on the other hand for non-smooth RCD * (K, N)-spaces it seems they have not yet appeared in the literature, and we expect them to be a useful technical tool in the future development of the eld.…”