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.     

A rough curvature-dimension condition for metric measure spaces

We introduce and study a rough (approximate) curvature-dimension condition for metric measure spaces, applicable especially in the framework of discrete spaces and graphs. This condition extends the one introduced by Karl-Theodor Sturm, in his 2006 article On the geometry of metric measure spaces II, to a larger class of (possibly non-geodesic) metric measure spaces. The rough curvature-dimension condition is stable under an appropriate notion of convergence, and stable under discretizations as well. For spaces that satisfy a rough curvature-dimension condition we prove a generalized Brunn-Minkowski inequality and a Bonnet-Myers type theorem.     

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.     

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.     

Some functional inequalities on non-reversible Finsler manifolds

We continue our study of geometric analysis on (possibly non-reversible) Finsler manifolds, based on the Bochner inequality established by Ohta and Sturm. Following the approach of the \(\Gamma \)-calculus of Bakry et al (2014), we show the dimensional versions of the Poincaré–Lichnerowicz inequality, the logarithmic Sobolev inequality, and the Sobolev inequality. In the reversible case, these inequalities were obtained by Cavalletti and Mondino (2015) in the framework of curvature-dimension condition by means of the localization method. We show that the same (sharp) estimates hold also for non-reversible metrics.     

(<Emphasis Type="Italic">K</Emphasis>, <Emphasis Type="Italic">N</Emphasis>)-Convexity and the Curvature-Dimension Condition for Negative <Emphasis Type="Italic">N</Emphasis>

We extend the range of N to negative values in the (K, N)-convexity (in the sense of Erbar–Kuwada–Sturm), the weighted Ricci curvature \(\mathop {\mathrm {Ric}}\nolimits _N\) and the curvature-dimension condition \(\mathop {\mathrm {CD}}\nolimits (K,N)\). We generalize a number of results in the case of \(N>0\) to this setting, including Bochner’s inequality, the Brunn–Minkowski inequality and the equivalence between \(\mathop {\mathrm {Ric}}\nolimits _N \ge K\) and \(\mathop {\mathrm {CD}}\nolimits (K,N)\). We also show an expansion bound for gradient flows of Lipschitz (K, N)-convex functions.     

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.     

Geometric inequalities on Heisenberg groups

We establish geometric inequalities in the sub-Riemannian setting of the Heisenberg group \(\mathbb H^n\). Our results include a natural sub-Riemannian version of the celebrated curvature-dimension condition of Lott–Villani and Sturm and also a geodesic version of the Borell–Brascamp–Lieb inequality akin to the one obtained by Cordero-Erausquin, McCann and Schmuckenschläger. The latter statement implies sub-Riemannian versions of the geodesic Prékopa–Leindler and Brunn–Minkowski inequalities. The proofs are based on optimal mass transportation and Riemannian approximation of \(\mathbb H^n\) developed by Ambrosio and Rigot. These results refute a general point of view, according to which no geometric inequalities can be derived by optimal mass transportation on singular spaces.     

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.     

On local Poincaré via transportation

It is shown that curvature-dimension bounds CD(N,K) for a metric measure space (X,d,m) in the sense of Sturm imply a weak L ¹-Poincaré-inequality provided (X,d) has m-almost surely no branching points. Work supported by the Alexander von Humboldt-Foundation (AvH).     

A curvature-dimension condition for metric measure spaces

We present a curvature-dimension condition $\mathit{CD}(K,N)$ for metric measure spaces $(M,\mathsf{d},m)$. 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 ${\mathrm{Ric}}_M(\xi ,\xi )?K?{|\xi |}^2$ for all $\xi \in \mathit{TM}$. 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 $\mathit{CD}(K,\infty )$.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).     

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.     

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.     

Sturm theory for the equation of vibrating beam

In this paper we develop an extension of the classical Sturm theory [C. Sturm, Sur une classe d'equations à derivée partielle, J. Math. Pures Appl. 1 (1836) 373-444], to study the oscillation properties for the eigenfunctions of some fourth-order two point boundary value problems on the interval [0,1]. We are mainly interested in the case when these problems have negative eigenvalues induced by the sign of the parameters in the boundary conditions. In particular, we give an asymptotic estimate of the number of zeros in (0,1) of the first eigenfunction in terms of the variation of parameters in the boundary conditions.     

On solvability of three spectra problem

Three spectral problems generated by the same Sturm–Liouville equation are considered: Neumann–Dirichlet problem (the Neumann condition at the left end and the Dirichlet condition at the right end) on the whole interval [0, a], Neumann–Dirichlet problem on and Dirichlet–Dirichlet problem on . The three spectra inverse problem, i.e. the problem of recovering the Sturm–Liuville equation using the three spectra of these boundary value problems is completely solved.     

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.     

Trace Formula for Sturm--Liouville Operators with Singular Potentials

Mathematical Notes - Suppose that u(x) is a function of bounded variation on the closed interval [0,π], continuous at the endpoints of this interval. Then the Sturm—Liouville operator...     

Inverse scattering problem for Sturm–Liouville operator with nonlinear dependence on the spectral parameter in the boundary condition

The inverse problem of the scattering theory for Sturm–Liouville operator on the half line with boundary condition depending quadratic on the spectral parameter is considered. Scattering data are defined, some properties of the scattering data are examined, the main equation is obtained, solvability of the integral equation is proved and uniqueness of algorithm to the potential with given scattering data is studied. Copyright © 2010 John Wiley & Sons, Ltd.     

On the Basic Property in $$L_p $$ of a System of Eigenfunctions of a Problem with the Square of the Spectral Parameter in the Boundary Condition

Differential Equations - We study the Sturm–Liouville problem with the square of the spectral parameter in one boundary condition, the other boundary condition being homogeneous. In the...     

Equiconvergence of Spectral Decompositions for Sturm–Liouville Operators with a Distributional Potential in Scales of Spaces

Doklady Mathematics - We study the equiconvergence of spectral decompositions for two Sturm–Liouville operators on the interval [0, π] generated by the differential expressions...