Curvature-dimension Condition Research Articles

Local Poincaré inequalities from stable curvature conditions on metric spaces

We prove local Poincaré inequalities under various curvature-dimension conditions which are stable under the measured Gromov–Hausdorff convergence. The first class of spaces we consider is that of weak CD(K, N) spaces as defined by Lott and Villani. The second class of spaces we study consists of spaces where we have a flow satisfying an evolution variational inequality for either the Rényi entropy functional $${\fancyscript{E}_{N}(\rho m) = -\int_{X} {\rho}^{1-1/N} dm}$$ or the Shannon entropy functional $${\fancyscript{E}_{\infty}({\rho} m) = \int_{X}{\rho}\log {\rho} dm.}$$ We also prove that if the Rényi entropy functional is strongly displacement convex in the Wasserstein space, then at every point of the space we have unique geodesics to almost all points of the space.

Equivalent semigroup properties for the curvature-dimension condition

Some equivalent gradient and Harnack inequalities of a diffusion semigroup are presented for the curvature-dimension condition of the associated generator. As applications, the first eigenvalue, the log-Harnack inequality, the heat kernel estimates, and the HWI inequality are derived by using the curvature-dimension condition. The transportation inequality for diffusion semigroups is also investigated.

Localization and tensorization properties of the curvature-dimension condition for metric measure spaces, II

This is an addendum to the paper [K. Bacher, K.T. Sturm, Localization and tensorization properties of the curvature-dimension condition for metric measure spaces, J. Funct. Anal. 259 (2010) 28–56]. We prove the tensorization property for the curvature-dimension condition, add some detailed calculations – including explicit dependence of constants – and comment on assumptions and conjectures concerning the local-to-global statement in Bacher and Sturm (2010) [1] and Villani (2009) [6], respectively.

Localization and tensorization properties of the curvature-dimension condition for metric measure spaces

This paper is devoted to the analysis of metric measure spaces satisfying locally the curvature-dimension condition CD ( K , N ) introduced by the second author and also studied by Lott & Villani. We prove that the local version of CD ( K , N ) is equivalent to a global condition CD ∗ ( K , N ) , slightly weaker than the (usual, global) curvature-dimension condition. This so-called reduced curvature-dimension condition CD ∗ ( K , N ) has the local-to-global property. We also prove the tensorization property for CD ∗ ( K , N ) . As an application we conclude that the fundamental group π 1 ( M , x 0 ) of a metric measure space ( M , d , m ) is finite whenever it satisfies locally the curvature-dimension condition CD ( K , N ) with positive K and finite N.

On Borell-Brascamp-Lieb Inequalities on Metric Measure Spaces

In this paper we introduce the notion of a Borell-Brascamp-Lieb inequality for metric measure spaces (M,d,m) denoted by BBL(K,N) for two numbers K,N ∈ ℝ with N ≥ 1. In the first part we prove that BBL(K,N) holds true on metric measure spaces satisfying a curvature-dimension condition CD(K,N) developed and studied by Lott and Villani in (Ann Math 169:903–991, 2007) as well as by Sturm in (Acta Math 196(1):133–177, 2006). The aim of the second part is to show that BBL(K,N) is stable under convergence of metric measure spaces with respect to the L2-transportation distance.

Finsler interpolation inequalities

We extend Cordero-Erausquin et al.’s Riemannian Borell–Brascamp–Lieb inequality to Finsler manifolds. Among applications, we establish the equivalence between Sturm, Lott and Villani’s curvature-dimension condition and a certain lower Ricci curvature bound. We also prove a new volume comparison theorem for Finsler manifolds which is of independent interest.

On the role of convexity in isoperimetry, spectral gap and concentration

We show that for convex domains in Euclidean space, Cheeger’s isoperimetric inequality, spectral gap of the Neumann Laplacian, exponential concentration of Lipschitz functions, and the a-priori weakest requirement that Lipschitz functions have arbitrarily slow uniform tail-decay, are all quantitatively equivalent (to within universal constants, independent of the dimension). This substantially extends previous results of Maz’ya, Cheeger, Gromov–Milman, Buser and Ledoux. As an application, we conclude a sharp quantitative stability result for the spectral gap of convex domains under convex perturbations which preserve volume (up to constants) and under maps which are “on-average” Lipschitz. We also provide a new characterization (up to constants) of the spectral gap of a convex domain, as one over the square of the average distance from the “worst” subset having half the measure of the domain. In addition, we easily recover and extend many previously known lower bounds on the spectral gap of convex domains, due to Payne–Weinberger, Li–Yau, Kannan–Lovász–Simonovits, Bobkov and Sodin. The proof involves estimates on the diffusion semi-group following Bakry–Ledoux and a result from Riemannian Geometry on the concavity of the isoperimetric profile. Our results extend to the more general setting of Riemannian manifolds with density which satisfy the CD(0,∞) curvature-dimension condition of Bakry-Émery.

On the role of convexity in functional and isoperimetric inequalities

This is a continuation of our previous work [Preprint, 2008, http://arxiv.org/abs/0712.4092]. It is well known that various isoperimetric inequalities imply their functional ‘counterparts’, but in general this is not an equivalence. We show that under certain convexity assumptions (for example, for log-concave probability measures in Euclidean space), the latter implication can in fact be reversed for very general inequalities, generalizing a reverse form of Cheeger's inequality due to Buser and Ledoux. We develop a coherent single framework for passing between isoperimetric inequalities, Orlicz–Sobolev functional inequalities and capacity inequalities, the latter being notions introduced by Maz'ya and extended by Barthe–Cattiaux–Roberto. As an application, we extend the known results due to the latter authors about the stability of the isoperimetric profile under tensorization, when there is no Central-Limit obstruction. As another application, we show that under our convexity assumptions, q-log-Sobolev inequalities (q ∈ [1, 2]) are equivalent to an appropriate family of isoperimetric inequalities, extending results of Bakry–Ledoux and Bobkov–Zegarliński. Our results extend to the more general setting of Riemannian manifolds with density which satisfy the CD(0, ∞) curvature–dimension condition of Bakry–Émery.

Uniform tail-decay of Lipschitz functions implies Cheeger's isoperimetric inequality under convexity assumptions

We show that for convex domains in Euclidean space, Cheeger's isoperimetric inequality, spectral-gap of the Neumann Laplacian, exponential concentration of Lipschitz functions, and the a priori weakest uniform tail-decay of these functions, are all equivalent (to within universal constants, independent of the dimension). This substantially extends previous results of Maz'ya, Cheeger, Gromov–Milman, Buser and Ledoux. As an application, we conclude the stability of the spectral-gap for convex domains under convex perturbations which preserve volume (up to constants) and under maps which are “on-average” Lipschitz. We also provide a new characterization of the Cheeger constant, as one over the expectation of the distance from the “worst” Borel set having half the measure of the convex domain. In addition, we easily recover (and extend) many previously known lower bounds, due to Payne–Weinberger, Li–Yau and Kannan–Lovász–Simonovits, on the Cheeger constant of convex domains. Essential to our proof is a result from Riemannian Geometry on the concavity of the isoperimetric profile. Our results extend to the more general setting of Riemannian manifolds with density which satisfy the CD ( 0 , ∞ ) curvature-dimension condition of Bakry–Émery. To cite this article: E. Milman, C. R. Acad. Sci. Paris, Ser. I 346 (2008).

Generalized Transportation-Cost Inequalities and Applications

For μ: = e V(x)dx a probability measure on a complete connected Riemannian manifold, we establish a correspondence between the Entropy-Information inequality $\Psi(\mu(f^2{\rm{log}} f^2))\le \mu(|{\nabla}\!\!f |^2)$ and the transportation-cost inequality $W_2(f^2\mu,\mu)\le \Phi(\mu(f^2{\rm{log}} f^2))$ for μ(f 2) = 1, where Φ and Ψ are increasing functions. Moreover, under the curvature–dimension condition, a Sobolev type HWI (entropy-cost-information) inequality is established. As applications, explicit estimates are obtained for the Sobolev constant and the diameter of a compact manifold, which either extend or improve some corresponding known results.

On the geometry of metric measure spaces. II

We introduce a curvature-dimension condition CD (K, N) for metric measure spaces. It is more restrictive than the curvature bound $\underline{{{\text{Curv}}}} {\left( {M,{\text{d}},m} \right)} > K$ (introduced in [Sturm K-T (2006) On the geometry of metric measure spaces. I. Acta Math 196:65–131]) which is recovered as the borderline case CD(K, ∞). The additional real parameter N plays the role of a generalized upper bound for the dimension. For Riemannian manifolds, CD(K, N) is equivalent to $${\text{Ric}}_{M} {\left( {\xi ,\xi } \right)} > K{\left| \xi \right|}^{2} $$ and dim(M) ⩽ N. The curvature-dimension condition CD(K, N) is stable under convergence. For any triple of real numbers K, N, L the family of normalized metric measure spaces (M, d, m) with CD(K, N) and diameter ⩽ L is compact. Condition CD(K, N) implies sharp version of the Brunn–Minkowski inequality, of the Bishop–Gromov volume comparison theorem and of the Bonnet–Myers theorem. Moreover, it implies the doubling property and local, scale-invariant Poincare inequalities on balls. In particular, it allows to construct canonical Dirichlet forms with Gaussian upper and lower bounds for the corresponding heat kernels.

A curvature-dimension condition for metric measure spaces

We present a curvature-dimension condition CD ( K , N ) for metric measure spaces ( M , d , m ) . In some sense, it will be the geometric counterpart to the Bakry–Émery [D. Bakry, M. Émery, Diffusions hypercontractives, in: Séminaire de Probabilités XIX, in: Lecture Notes in Math., vol. 1123, Springer, Berlin, 1985, pp. 177–206. [1]] condition for Dirichlet forms. For Riemannian manifolds, it holds if and only if dim ( M ) ⩽ N and Ric M ( ξ , ξ ) ⩾ K ⋅ | ξ | 2 for all ξ ∈ TM . The curvature bound introduced in [J. Lott, C. Villani, Ricci curvature for metric-measure spaces via optimal transport, Annals of Math., in press. [4]; K.T. Sturm, Generalized Ricci bounds and convergence of metric measure spaces, C. R. Acad. Sci. Paris, Ser. I 340 (2005) 235–238. [6]; K.T. Sturm, On the geometry of metric measure spaces. I, Acta Math., in press. [7]] is the limit case CD ( K , ∞ ) . Our curvature-dimension condition is stable under convergence. Furthermore, it entails various geometric consequences e.g. the Bishop–Gromov theorem and the Bonnet–Myers theorem. In both cases, we obtain the sharp estimates known from the Riemannian case. To cite this article: K.-T. Sturm, C. R. Acad. Sci. Paris, Ser. I 342 (2006).

Heat Kernel Estimates with Application to Compactness of Manifolds

Li and Yau type two‐sided heat kernel bounds are obtained for symmetric diffusions under a curvature–dimension condition, where the heat kernel upper bound is established for a more general case. As an application, the compactness of manifolds is studied using heat kernels. In particular, a conjecture by Bueler is proved.

Graphs with large girth and nonnegative curvature dimension condition

In this paper, we classify unweighted graphs satisfying the curvature dimension condition CD(0,\infty) whose girth are at least five.