On the Geometry of Metric Measure Spaces with Variable Curvature Bounds

Motivated by a classical comparison result of J. C. F. Sturm, we introduce a curvature-dimension condition CD(k, N) for general metric measure spaces, variable lower curvature bound \(k\) and upper dimension bound \(N\ge 1\). In the case of non-zero constant lower curvature, our approach coincides with the celebrated condition that was proposed by Sturm (Acta Math 196(1):133–177, 2006). We prove several geometric properties as sharp Bishop–Gromov volume growth comparison or a sharp generalized Bonnet–Myers theorem (Schneider’s Theorem). In addition, the curvature-dimension condition is stable with respect to measured Gromov–Hausdorff convergence, and it is stable with respect to tensorization of finitely many metric measure spaces provided a non-branching condition is assumed. We also briefly describe possible extensions for variable dimension bounds.     

Concentration, Ricci Curvature, and Eigenvalues of Laplacian

In this paper we study the concentration behavior of metric measure spaces. We prove the stability of the curvature-dimension condition with respect to the concentration topology due to Gromov. As an application, under the nonnegativity of Bakry–Émery Ricci curvature, we prove that the kth eigenvalue of the weighted Laplacian of a closed Riemannian manifold is dominated by a constant multiple of the first eigenvalue, where the constant depends only on k and is independent of the dimension of the manifold.     

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 π₁(M,x₀) 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.     

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.     

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 L ₂-transportation distance.     

Mass transportation and rough curvature bounds for discrete spaces

We introduce and study rough (approximate) lower curvature bounds for discrete spaces and for graphs. This notion agrees with the one introduced in [J. Lott, C. Villani, Ricci curvature for metric-measure spaces via optimal transport, Ann. of Math. 169 (2009), in press] and [K.T. Sturm, On the geometry of metric measure spaces. I, Acta Math. 196 (2006) 65-131], in the sense that the metric measure space which is approximated by a sequence of discrete spaces with rough curvature ?K will have curvature ?K in the sense of [J. Lott, C. Villani, Ricci curvature for metric-measure spaces via optimal transport, Ann. of Math. 169 (2009), in press; K.T. Sturm, On the geometry of metric measure spaces. I, Acta Math. 196 (2006) 65-131]. Moreover, in the converse direction, discretizations of metric measure spaces with curvature ?K will have rough curvature ?K. We apply our results to concrete examples of homogeneous planar graphs.     

Conformal Transformation on Metric Measure Spaces

Potential Analysis - We study several problems concerning conformal transformation on metric measure spaces, including the Sobolev space, the differential structure and the curvature-dimension...     

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 (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 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 Poincaré inequalities on balls. In particular, it allows to construct canonical Dirichlet forms with Gaussian upper and lower bounds for the corresponding heat kernels.     

Non-embedding theorems of nilpotent Lie groups and sub-Riemannian manifolds

Wo prove that there do not exist quasi-isometric embeddings of connected nonabelian nilpotent Lie groups equipped with left invariant Riemannian metrics into a metric measure space satisfying the curvature-dimension condition RCD(Q,N)with N∈R and N>1.In fact,we can prove that a sub-Riemannian manifold whose generic degree of nonholonomy is not smaller than 2 cannot be bi-Lipschitzly embedded in any Banach space with the Radon-Nikodym property.We also get that every regular sub-Riemannian manifold do not satisfy the curvature-dimension condition CD(K,N),where K,N∈R and N>1.Along the way to the proofs,we show that the minimal weak upper gradient and the horizontal gradient coincide on the Carnot-Caratheodory spaces which may have independent interests.     

Quantitative Isoperimetry à la Levy-Gromov

On a Riemannian manifold with a positive lower bound on the Ricci tensor, the distance of isoperimetric sets from geodesic balls is quantitatively controlled in terms of the gap between the isoperimetric profile of the manifold and that of a round sphere of suitable radius. The deficit between the diameters of the manifold and of the corresponding sphere is likewise bounded. These results are actually obtained in the more general context of (possibly nonsmooth) metric measure spaces with curvature-dimension conditions through a quantitative analysis of the transport rays decompositions obtained by the localization method. © 2018 Wiley Periodicals, Inc.     

Strong metric subregularity of mappings in variational analysis and optimization

Although the property of strong metric subregularity of set-valued mappings has been present in the literature under various names and with various (equivalent) definitions for more than two decades, it has attracted much less attention than its older “siblings”, the metric regularity and the strong (metric) regularity. The purpose of this paper is to show that the strong metric subregularity shares the main features of these two most popular regularity properties and is not less instrumental in applications. We show that the strong metric subregularity of a mapping F acting between metric spaces is stable under perturbations of the form $f+F$, where f is a function with a small calmness constant. This result is parallel to the Lyusternik–Graves theorem for metric regularity and to the Robinson theorem for strong regularity, where the perturbations are represented by a function f with a small Lipschitz constant. Then we study perturbation stability of the same kind for mappings acting between Banach spaces, where f is not necessarily differentiable but admits a set-valued derivative-like approximation. Strong metric q-subregularity is also considered, where q is a positive real constant appearing as exponent in the definition. Rockafellar's criterion for strong metric subregularity involving injectivity of the graphical derivative is extended to mappings acting in infinite-dimensional spaces. A sufficient condition for strong metric subregularity is established in terms of surjectivity of the Fréchet coderivative, and it is shown by a counterexample that surjectivity of the limiting coderivative is not a sufficient condition for this property, in general. Then various versions of Newton's method for solving generalized equations are considered including inexact and semismooth methods, for which superlinear convergence is shown under strong metric subregularity. As applications to optimization, a characterization of the strong metric subregularity of the KKT mapping is obtained, as well as a radius theorem for the optimality mapping of a nonlinear programming problem. Finally, an error estimate is derived for a discrete approximation in optimal control under strong metric subregularity of the mapping involved in the Pontryagin principle.     

A differentiable structure for metric measure spaces

The main result of this paper is the provision of conditions under which a metric measure space admits a differentiable structure. This differentiable structure gives rise to a finite-dimensional L^∞ cotangent bundle over the given metric measure space and then to a Sobolev space H^1,p over the given metric measure space, the latter which is reflexive for p>1. This extends results of Cheeger (Geom. Funct. Anal. 9 (1999) (3) 428) to a wider collection of metric measure spaces.     

Monotonicity of Time-Dependent Transportation Costs and Coupling by Reflection

Based on a study of the coupling by reflection of diffusion processes, a new monotonicity in time of a time-dependent transportation cost between heat distribution is shown under Bakry-Émery’s curvature-dimension condition on a Riemannian manifold. The cost function comes from the total variation between heat distributions on spaceforms. As a corollary, we obtain a comparison theorem for the total variation between heat distributions. In addition, we show that our monotonicity is stable under the Gromov-Hausdorff convergence of the underlying space under a uniform curvature-dimension and diameter bound.     

Differentiability of p-Harmonic Functions on Metric Measure Spaces

We study p-harmonic functions on metric measure spaces, which are formulated as minimizers to certain energy functionals. For spaces supporting a p-Poincaré inequality, we show that such functions satisfy an infinitesmal Lipschitz condition almost everywhere. This result is essentially sharp, since there are examples of metric spaces and p-harmonic functions that fail to be locally Lipschitz continuous on them. As a consequence of our main theorem, we show that p-harmonic functions also satisfy a generalized differentiability property almost everywhere, in the sense of Cheeger’s measurable differentiable structures.     

Fixed point results for mappings satisfying (ψ φ)-weakly contractive condition in ordered partial metric spaces

In [18], Matthews introduced a new class of metric spaces, that is, the concept of partial metric spaces, or equivalently, weightable quasi-metrics, are investigated to generalize metric spaces (X, d), to develop and to introduce a new fixed point theory. In partial metric spaces, the self-distance for any point need not be equal to zero. In this paper, we study some results for single map satisfying (ψ,φ)-weakly contractive condition in partial metric spaces endowed with partial order. An example is given to support the useability of our results.     

Two valued measure and summability of double sequences in asymmetric context

We extend the ideas of convergence and Cauchy condition of double sequences extended by a two valued measure (called ??-statistical convergence/Cauchy condition and convergence/Cauchy condition in ??-density, studied for real numbers in our recent paper [7]) to a very general structure like an asymmetric (quasi) metric space. In this context it should be noted that the above convergence ideas naturally extend the idea of statistical convergence of double sequences studied by Móricz [15] and Mursaleen and Edely [17]. We also apply the same methods to introduce, for the first time, certain ideas of divergence of double sequences in these abstract spaces. The asymmetry (or rather, absence of symmetry) of asymmetric metric spaces not only makes the whole treatment different from the real case [7] but at the same time, like [3], shows that symmetry is not essential for any result of [7] and in certain cases to get the results, we can replace symmetry by a genuinely asymmetric condition called (AMA).     

Martin Kernels for Markov Processes with Jumps

We prove the existence of boundary limits of ratios of positive harmonic functions for a wide class of Markov processes with jumps and irregular (possibly disconnected) domains of harmonicity, in the context of general metric measure spaces. As a corollary, we prove the uniqueness of the Martin kernel at each boundary point, that is, we identify the Martin boundary with the topological boundary. We also prove a Martin representation theorem for harmonic functions. Examples covered by our results include: strictly stable Lévy processes in R ^d with positive continuous density of the Lévy measure; stable-like processes in R ^d and in domains; and stable-like subordinate diffusions in metric measure spaces.     

Orlicz-Sobolev extensions and measure density condition

We study the extension properties of Orlicz-Sobolev functions both in Euclidean spaces and in metric measure spaces equipped with a doubling measure. We show that a set E⊂R satisfying a measure density condition admits a bounded linear extension operator from the trace space W1,Ψ(Rn)E_| to W1,Ψ(Rn). Then we show that a domain, in which the Sobolev embedding theorem or a Poincaré-type inequality holds, satisfies the measure density condition. It follows that the existence of a bounded, possibly non-linear extension operator or even the surjectivity of the trace operator implies the measure density condition and hence the existence of a bounded linear extension operator.     

The Stepanov differentiability theorem in metric measure spaces

We extend Cheeger’s theorem on differentiability of Lipschitz functions in metric measure spaces to the class of functions satisfying Stepanov’s condition. As a consequence, we obtain the analogue of Calderon’s differentiability theorem of Sobolev functions in metric measure spaces satisfying a Poincaré inequality. Communicated by Steven Krantz