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Ricci curvature, entropy and optimal transport
Abstract
This chapter comprises the lecture notes on the interplay between optimal transport and Riemannian geometry. On a Riemannian manifold, the convexity of entropy along optimal transport in the space of probability measures characterizes lower bounds of the Ricci curvature. We then discuss geometric properties of general metric measure spaces satisfying this convexity condition. Introduction This chapter is extended notes based on the author's lecture series at the summer school at Université Joseph Fourier, Grenoble: “Optimal Transportation: Theory and Applications.” The aim of these five lectures (corresponding to Sections 7.3–7.7) was to review the recent impressive development on the interplay between optimal transport theory and Riemannian geometry. Ricci curvature and entropy are the key ingredients. See [Lo2] for a survey in the same spirit with a slightly different selection of topics. Optimal transport theory is concerned with the behavior of transport between two probability measures in a metric space. We say that such transport is optimal if it minimizes a certain cost function typically defined from the distance of the metric space. Optimal transport naturally inherits the geometric structure of the underlying space; in particular Ricci curvature plays a crucial role for describing optimal transport in Riemannian manifolds. In fact, optimal transport is always performed along geodesics, and we obtain Jacobi fields as their variational vector fields. The behavior of these Jacobi fields is controlled by the Ricci curvature as is usual in comparison geometry. In this way, a lower Ricci curvature bound turns out to be equivalent to a certain convexity property of entropy in the space of probability measures.
... We conclude this brief review with a statement of the famous Bishop-Gromov Comparison Theorem. This is well known and can be found in any book on comparison geometry; we refer to the notes in [29] for an excellent introduction to the basic theory at an elementary level: Theorem 4 (Bishop-Gromov). Let (M, g) a d -dimensional Riemannian manifold such that the scalar curvature is bounded below by −K for some K > 0. For any point x ∈ M , let B r (x) denote the metric ball around x in M with radius r > 0. For any 0 < r < R, we have ...
... Proofs of both results are available in the third section of [29]. ...
Recently, the task of image generation has attracted much attention. In particular, the recent empirical successes of the Markov Chain Monte Carlo (MCMC) technique of Langevin Dynamics have prompted a number of theoretical advances; despite this, several outstanding problems remain. First, the Langevin Dynamics is run in very high dimension on a nonconvex landscape; in the worst case, due to the NP-hardness of nonconvex optimization, it is thought that Langevin Dynamics mixes only in time exponential in the dimension. In this work, we demonstrate how the manifold hypothesis allows for the considerable reduction of mixing time, from exponential in the ambient dimension to depending only on the (much smaller) intrinsic dimension of the data. Second, the high dimension of the sampling space significantly hurts the performance of Langevin Dynamics; we leverage a multi-scale approach to help ameliorate this issue and observe that this multi-resolution algorithm allows for a trade-off between image quality and computational expense in generation.
... Remark 2.18 (Curvature-dimension condition) The original motivation of introducing Ric N in [Oh3] was to study Lott, Sturm and Villani's curvature-dimension condition CD(K, N) on Finsler manifolds (see also the surveys [Oh4,Oh6]). The celebrated condition CD(K, N) was introduced as a synthetic geometric notion playing a role of a lower Ricci curvature bound for metric measure spaces ( [St1,St2,LV]). ...
This is a survey article on recent progress of comparison geometry and geometric analysis on Finsler manifolds of weighted Ricci curvature bounded below. Our purpose is two-fold: Give a concise and geometric review on the birth of weighted Ricci curvature and its applications; Explain recent results from a nonlinear analogue of the -calculus based on the Bochner inequality. In the latter we discuss some gradient estimates, functional inequalities, and isoperimetric inequalities.
... We briefly review the basics of Finsler geometry (we refer to [BCS, Sh, SS] for further reading), and some facts from [Oh2,OS1,OS3]. Interested readers can consult [Oh7] for more elaborate preliminaries, as well as related surveys [Oh3,Oh4,Oh8]. ...
We continue our study of geometric analysis on (possibly non-reversible) Finsler manifolds, based on the Bochner inequality established by the author and Sturm. Following the approach of the -calculus a la Bakry et al, we show the dimensional versions of the Poincare--Lichnerowicz inequality, the logarithmic Sobolev inequality, and the Sobolev inequality. In the reversible case, these inequalities were obtained by Cavalletti--Mondino 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.
... This theory is making a breathtaking progress in this decade. We refer to the book [Vi2] for a detailed account at that time (2009), and to [Oh6] for a survey from a more geometric viewpoint. ...
Klartag recently gave a beautiful alternative proof of the isoperimetric inequalities of Levy-Gromov, Bakry-Ledoux, Bayle and E. Milman on weighted Riemannian manifolds. Klartag's approach is based on a generalization of the localization method (so-called needle decompositions) in convex geometry, inspired also by optimal transport theory. Cavalletti and Mondino subsequently generalized the localization method, in a different way more directly along optimal transport theory, to essentially non-branching metric measure spaces satisfying the curvature-dimension condition. This class in particular includes reversible (absolutely homogeneous) Finsler manifolds. In this paper, we construct needle decompositions of non-reversible (only positively homogeneous) Finsler manifolds, and show an isoperimetric inequality under bounded reversibility constants. A discussion on the curvature-dimension condition CD(K,N) for is also included, it would be of independent interest.
... The original motivation of introducing Ric N in [44] was to study Lott, Sturm and Villani's curvature-dimension condition CD(K , N ) on Finsler manifolds (see also the surveys [46,49]). The celebrated condition CD(K , N ) was introduced as a synthetic geometric notion playing a role of a lower Ricci curvature bound for metric measure spaces [36,67,68]. ...
This is a survey article on recent progress of comparison geometry and geometric analysis on Finsler manifolds of weighted Ricci curvature bounded below. Our purpose is twofold: give a concise and geometric review on the birth of weighted Ricci curvature and its applications; explain recent results from a nonlinear analogue of the Γ-calculus based on the Bochner inequality. In the latter we discuss some gradient estimates, functional inequalities, and isoperimetric inequalities.
... We briefly review the basics of Finsler geometry (we refer to [BCS, Sh, SS] for further reading), and some facts from [Oh2,OS1,OS3]. Interested readers can consult [Oh7] for more elaborate preliminaries, as well as related surveys [Oh3,Oh4,Oh8]. Throughout the article, let M be a connected, n-dimensional C ∞ -manifold without boundary such that n ≥ 2. We also fix an arbitrary positive C ∞ -measure m on M. ...
We continue our study of geometric analysis on (possibly non-reversible) Finsler manifolds, based on the Bochner inequality established by the author and Sturm. Following the approach of the -calculus a la Bakry et al, we show the dimensional versions of the Poincare--Lichnerowicz inequality, the logarithmic Sobolev inequality, and the Sobolev inequality. In the reversible case, these inequalities were obtained by Cavalletti--Mondino 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.
... For Ricci curvature, promising attempts have been made only recently by Lott and Villani in [38], and by Sturm in [59,60]. Both use optimal transport as well as a suitable abstraction of volume comparison (see [43] for a comprehensive introduction). Also in the case of a Lorentzian manifold, one can of course ask for synthetic curvature bounds. ...
The present thesis is concerned with geometric consequences of a lower bound on the Ricci curvature of a Riemannian or a Lorentzian manifold.
In the Riemannian case, a detailed exposition of the classical Bishop-Gromov comparison result for the volume of geodesic balls is presented. Moreover, the well-known relation between positive Ricci curvature and finite diameter (Myers's theorem) is derived as well. Here the more classical approach via the index form and Jacobi fields is compared to the rather modern approach using volume comparison.
In the Lorentzian case, the aim was to adopt techniques from Riemannian geometry to obtain similar comparison results also for Lorentzian manifolds. In particular, it was desirable to obtain a global volume comparison result similar to the Bishop-Gromov theorem without any restrictions to "small" neighborhoods. One of the key points in order to do so is to obtain a good understanding of (differentiability) properties of the Lorentzian time separation function. This is achieved by a detailed study of causal properties and the cut locus of a Lorentzian manifold. For globally hyperbolic Lorentzian manifolds, this indeed yields sufficient properties in order to obtain global volume comparison results analogous to the Bishop-Gromov theorem. Moreover, as a consequence of these comparison results, we derive a so-called singularity theorem, i.e. a statement about geodesic incompleteness of a Lorentzian manifold. Specifically, we recover a version of Hawking's singularity theorem.
Along the way, we also discuss comparison results for the Laplacian of the Riemannian distance function, for the d'Alembertian of the Lorentzian time separation, and for areas of geodesic spheres. At least in the Riemannian case these results are well-known.
... See [St2, Proposition 2.1, Theorem 2.3] (and[Vi, Theorem 30.7],[Oh3, Theorem 6.1] as well for the case of N = ∞) for the precise statements of the Brunn-Minkowski inequality and the Bishop-Gromov volume comparison. ...
We consider Hilbert and Funk geometries on a strongly convex domain in the
Euclidean space. We show that, with respect to the Lebesgue measure on the
domain, Hilbert (resp. Funk) metric has the bounded (resp. constant negative)
weighted Ricci curvature. As one of corollaries, these metric measure spaces
satisfy the curvature-dimension condition in the sense of Lott, Sturm and
Villani.
In Part I, we saw that the natural notions of Finsler curvatures (the flag and Ricci curvatures) can be introduced through the behavior of geodesics, and then several comparison theorems follow smoothly by similar arguments to the Riemannian case, or through the characterizations of these curvatures from the Riemannian geometric point of view. In order to proceed further in this direction, we would like to equip our Finsler manifold with a measure on it. At this point, however, we face a difficulty in choosing a measure, because a Finsler manifold does not necessarily have a unique canonical measure like the volume measure in the Riemannian case. Then our standpoint is that, instead of choosing some constructive measure, we begin with an arbitrary measure and modify the Ricci curvature into the weighted Ricci curvature according to the choice of a measure. This is motivated by the theory deeply investigated in the Riemannian case by Lichnerowicz, Bakry and others. It will turn out that this strategy fits the Finsler setting very well.
Since the end of twentieth century, optimal transport theory has been making a breathtaking and diverge progress in and outside mathematics, e.g., partial differential equations, probability theory, differential geometry, economics, and image processing, to name a few. What is especially relevant to our interest is the convexity of an entropy functional along optimal transports, called the curvature-dimension condition. This notion, due to Lott, Sturm, and Villani, turned out having rich applications in analysis and geometry. In this chapter, in our framework of Finsler manifolds, we overview the basic ideas of optimal transport theory and its relation with the weighted Ricci curvature.
Klartag recently gave a beautiful alternative proof of the isoperimetric
inequalities of Levy-Gromov, Bakry-Ledoux and E. Milman on weighted Riemannian
manifolds. Klartag's approach is based on needle decompositions associated with
1-Lipschitz functions, inspired by convex geometry and optimal transport
theory. Cavalletti and Mondino subsequently generalized Klartag's technique to
essentially non-branching metric measure spaces satisfying the
curvature-dimension condition, in particular including reversible (absolutely
homogeneous) Finsler manifolds. In this paper, we construct needle
decompositions of non-reversible (only positively homogeneous) Finsler
manifolds, and show an isoperimetric inequality under bounded reversibility
constants. A discussion on the curvature-dimension condition CD(K,N) (in the
sense of Lott-Sturm-Villani) for N=0 is also included, it would be of
independent interest.
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.
We investigate the distribution of eigenvalues of the weighted Laplacian on
closed weighted Riemannian manifolds of nonnegative Bakry-\'Emery Ricci
curvature. We derive some universal inequalities among eigenvalues of the
weighted Laplacian on such manifolds. These inequalities are quantitative
versions of the previous theorem by the author with Shioya. We also study some
geometric quantity, called multi-way isoperimetric constants, on such manifolds
and obtain similar universal inequalities among them. Multi-way isoperimetric
constants are generalizations of the Cheeger constant. Extending and following
the heat semigroup argument by Ledoux and E. Milman, we extend the Buser-Ledoux
result to the k-th eigenvalue and the k-way isoperimetric constant. As a
consequence the k-th eigenvalue of the weighted Laplacian and the k-way
isoperimetric constant are equivalent up to polynomials of k on closed
weighted manifolds of nonnegative Bakry-\'Emery Ricci curvature.
A concavity estimate is derived for interpolations between L
1(M) mass densities on a Riemannian manifold. The inequality sheds new light on the theorems of Prékopa, Leindler, Borell, Brascamp
and Lieb that it generalizes from Euclidean space. Due to the curvature of the manifold, the new Riemannian versions of these
theorems incorporate a volume distortion factor which can, however, be controlled via lower bounds on Ricci curvature. The
method uses optimal mappings from mass transportation theory. Along the way, several new properties are established for optimal
mass transport and interpolating maps on a Riemannian manifold.
We introduce and analyze lower (Ricci) curvature bounds
\underline{{Curv}} {\left( {M,d,m} \right)}
⩾ K for metric measure spaces
{\left( {M,d,m} \right)}
. Our definition is based on convexity properties of the relative entropy
Ent{\left( { \cdot \left| m \right.} \right)}
regarded as a function on the L
2-Wasserstein space of probability measures on the metric space
{\left( {M,d} \right)}
. Among others, we show that
\underline{{Curv}} {\left( {M,d,m} \right)}
⩾ K implies estimates for the volume growth of concentric balls. For Riemannian manifolds,
\underline{{Curv}} {\left( {M,d,m} \right)}
⩾ K if and only if
Ric_{M} {\left( {\xi ,\xi } \right)}
⩾ K
{\left| \xi \right|}^{2}
for all
\xi \in TM
.
The crucial point is that our lower curvature bounds are stable under an appropriate notion of D-convergence of metric measure spaces. We define a complete and separable length metric D on the family of all isomorphism classes of normalized metric measure spaces. The metric D has a natural interpretation, based on the concept of optimal mass transportation.
We also prove that the family of normalized metric measure spaces with doubling constant ⩽ C is closed under D-convergence. Moreover, the family of normalized metric measure spaces with doubling constant ⩽ C and diameter ⩽ L is compact under D-convergence.
Let (M,g) be a connected compact manifold, C3 smooth and without boundary, equipped with a Riemannian distance d(x,y). If s : M ® M s : M \to M is merely Borel and never maps positive volume into zero volume, we show s = t °u s = t \circ u factors uniquely a.e. into the composition of a map t(x) = expx[-Ñy(x)] t(x) = {\rm exp}_x[-\nabla\psi(x)] and a volume-preserving map u : M ® M u : M \to M , where
y: M ® \bold R \psi : M \to {\bold R} satisfies the additional property that (yc)c = y (\psi^c)^c = \psi with yc(y) :=inf{c(x,y) - y(x)|x Î M} \psi^c(y) :={\rm inf}\{c(x,y) - \psi(x)\,\vert\,x \in M\} and c(x,y) = d2(x,y)/2. Like the factorization it generalizes from Euclidean space, this non-linear decomposition can be linearized around the identity to yield the Hodge decomposition of vector fields.The results are obtained by solving a Riemannian version of the Monge--Kantorovich problem, which means minimizing the expected value of the cost c(x,y) for transporting one distribution f ³ 0 f \ge 0 of mass in L1(M) onto another. Parallel results for other strictly convex cost functions c(x,y) ³ 0 c(x,y) \ge 0 of the Riemannian distance on non-compact manifolds are briefly discussed.
A new set of inequalities is introduced, based on a novel but natural interpolation between Borel probability measures onRd. Using these estimates in lieu of convexity or rearrangement inequalities, the existence and uniqueness problems are solved for a family of attracting gas models. In these models, the gas interacts with itself through a force which increases with distance and is governed by an equation of stateP=P(ϱ) relating pressure to density.P(ϱ)/ϱ>(d−1)/dis assumed non-decreasing for ad-dimensional gas. By showing that the internal and potential energies for the system are convex functions of the interpolation parameter, an energy minimizing state—unique up to translation—is proven to exist. The concavity established for ¶ρt¶−p/dqas a function oft∈[0, 1] generalizes the Brunn–Minkowski inequality from sets to measures.
The Bakry-Emery tensor gives an analog of the Ricci tensor for a Riemannian manifold with a smooth measure. We show that some of the topological consequences of having a positive or nonnegative Ricci tensor are also valid for the Bakry-Emery tensor. We show that the Bakry-Emery tensor is nondecreasing under a Riemannian submersion whose fiber transport preserves measures up to constants. We give some relations between the Bakry-Emery tensor and measured Gromov-Hausdorff limits.
Jury : Dominique BAKRY (Professeur, Université Paul Sabatier), Rapporteur; Gérard BESSON (CNRS, Université de Grenoble I), Directeur de thèse; Pierre BERARD (Professeur, Université de Grenoble I); Thierry COULHON (Professeur, Université de Cergy-Pontoise), Président Sylvestre GALLOT (Professeur, Université de Grenoble I); Hervé PAJOT (Professeur, Université de Grenoble I); Antonio ROS (Professeur, Université de Grenade), Rapporteur hors jury
We investigate Prékopa-Leindler type inequalities on a Riemannian manifold M equipped with a measure with density e -V where the potential V and the Ricci curvature satisfy Hess x V+Ric x ≥λI for all x∈M, with some λ∈ℝ. As in our earlier work [the authors, Invent. Math. 146, No. 2, 219–257 (2001; Zbl 1026.58018)], the argument uses optimal mass transport on M, but here, with a special emphasis on its connection with Jacobi fields. A key role will be played by the differential equation satisfied by the determinant of a matrix of Jacobi fields. We also present applications of the method to logarithmic Sobolev inequalities (the Bakry-Emery criterion will be recovered) and to transport inequalities. A study of the displacement convexity of the entropy functional completes the exposition.
This book provides an introduction to Riemannian geometry, the geometry of curved spaces, for use in a graduate course. Requiring only an understanding of differentiable manifolds, the author covers the introductory ideas of Riemannian geometry followed by a selection of more specialized topics. Also featured are Notes and Exercises for each chapter, to develop and enrich the reader's appreciation of the subject. This second edition, first published in 2006, has a clearer treatment of many topics than the first edition, with new proofs of some theorems and a new chapter on the Riemannian geometry of surfaces. The main themes here are the effect of the curvature on the usual notions of classical Euclidean geometry, and the new notions and ideas motivated by curvature itself. Completely new themes created by curvature include the classical Rauch comparison theorem and its consequences in geometry and topology, and the interaction of microscopic behavior of the geometry with the macroscopic structure of the space.
We introduce a curvature-dimension condition CD (K, N) for metric measure spaces. It is more restrictive than the curvature bound
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 Poincar inequalities on balls. In particular,
it allows to construct canonical Dirichlet forms with Gaussian upper and lower bounds for the corresponding heat kernels.
CONTENTS § 1. Introduction § 2. Basic concepts § 3. Globalization theorem § 4. Natural constructions § 5. Burst points § 6. Dimension § 7. The tangent cone and the space of directions. Conventions and notation § 8. Estimates of rough volume and the compactness theorem § 9. Theorem on almost isometry §10. Hausdorff measure §11. Functions that have directional derivatives, the method of successive approximations, level surfaces of almost regular maps §12. Level lines of almost regular maps §13. Subsequent results and open questionsReferences
We present some new results concerning well-posedness of gradient flows generated by λ-convex functionals in a wide class of metric spaces, including Alexandrov spaces satisfying a lower curvature bound and the corresponding L2-Wasserstein spaces. Applications to the gradient flow of Entropy functionals in metric-measure spaces with Ricci curvature bounded from below and to the corresponding diffusion semigroup are also considered. These results have been announced during the workshop on “Optimal Transport: theory and applications” held in Pisa, November 2006. To cite this article: G. Savaré, C. R. Acad. Sci. Paris, Ser. I 345 (2007).
Introduction The Kantorovich duality Geometry of optimal transportation Brenier's polar factorization theorem The Monge-Ampere equation Displacement interpolation and displacement convexity Geometric and Gaussian inequalities The metric side of optimal transportation A differential point of view on optimal transportation Entropy production and transportation inequalities Problems Bibliography Table of short statements Index.
Singular spaces with upper curvature bounds and, in particular, spaces of nonpositive curvature, have been of interest in many fields, including geometric (and combinatorial) group theory, topology, dynamical systems and probability theory. In the first two chapters of the book, a concise introduction into these spaces is given, culminating in the Hadamard-Cartan theorem and the discussion of the ideal boundary at infinity for simply connected complete spaces of nonpositive curvature. In the third chapter, qualitative properties of the geodesic flow on geodesically complete spaces of nonpositive curvature are discussed, as are random walks on groups of isometries of nonpositively curved spaces. The main class of spaces considered should be precisely complementary to symmetric spaces of higher rank and Euclidean buildings of dimension at least two (Rank Rigidity conjecture). In the smooth case, this is known and is the content of the Rank Rigidity theorem. An updated version of the proof of the latter theorem (in the smooth case) is presented in Chapter IV of the book. This chapter contains also a short introduction into the geometry of the unit tangent bundle of a Riemannian manifold and the basic facts about the geodesic flow. In an appendix by Misha Brin, a self-contained and short proof of the ergodicity of the geodesic flow of a compact Riemannian manifold of negative curvature is given. The proof is elementary and should be accessible to the non-specialist. Some of the essential features and problems of the ergodic theory of smooth dynamical systems are discussed, and the appendix can serve as an introduction into this theory.
Let X be an n -dimensional Alexandrov space of curvature bounded from below. We define the notion of singular point in X, and prove that the set Sχ of singular points in X is of Hausdorff dimension ≤ n - 1 and that X - Sx has a natural C°-Riemannian structure. © 1994, International Press of Boston, Inc. All Rights Reserved.
We do three things. First, we characterize the class of measures µ ∈ P 2 (M) such that for any other ν ∈ P 2 (M) there exists a unique optimal transport plan, and this plan is induced by a map. Second, we study the tangent space at any measure and we identify the class of measures for which the tangent space is an Hilbert space. Third, we prove that these two classes of measures coincide. This answers a question recently raised by Villani. Our results concerning the tangent space can be extended to the case of Alexandrov spaces.
We present various characterizations of uniform lower bounds for the Ricci curvature of a smooth Riemannian manifold M in terms of convexity properties of the entropy (considered as a function on the space of probability measures on M) as well as in terms of transportation inequalities for volume measures, heat kernels and Brownian motions and in terms of gradient estimates for the heat semigroup.
This is a survey article on recent progress (in [Oh3], [OS]) of the theory of weighted Ricci curvature in Finsler geometry. Optimal trans-port theory plays an impressive role as is developed in the Riemannian case by Lott, Sturm and Villani.
Given a probability space (X, μ) and a bounded domain Ω in ℝd equipped with the Lebesgue measure |·| (normalized so that |Ω| = 1), it is shown (under additional technical assumptions on X and Ω) that for every vector-valued function u ∈ Lp (X, μ; ℝd) there is a unique “polar factorization” u = ∇Ψs, where Ψ is a convex function defined on Ω and s is a measure-preserving mapping from (X, μ) into (Ω, |·|), provided that u is nondegenerate, in the sense that μ(u−1(E)) = 0 for each Lebesgue negligible subset E of ℝd.
Through this result, the concepts of polar factorization of real matrices, Helmholtz decomposition of vector fields, and nondecreasing rearrangements of real-valued functions are unified.
The Monge-Ampère equation is involved in the polar factorization and the proof relies on the study of an appropriate “Monge-Kantorovich” problem.
We prove that iterated spaces of directions of a limit of a noncollapsing sequence of manifolds with lower curvature bound
are topologically spheres. As an application we show that for any finite dimensional Alexandrov space X
n
with there exists an Alexandrov space Y homeomorphic to X which cannot be obtained as such a limit.
We prove several sharp inequalities specifying the uniform convexity and uniform smoothness properties of the Schatten trace ideals C
p, which are the analogs of the Lebesgue spaces L
p in non-commutative integration. The inequalities are all precise analogs of results which had been known in L
p, but were only known in C
p for special values of p. In the course of our treatment of uniform convexity and smoothness inequalities for C
p we obtain new and simple proofs of the known inequalities for L
p.
In this paper we construct a "synthetic" parallel transportation along a geodesic in Alexandrov space with curvature bounded
below, and prove an analog of the second variation formula for this case. A closely related construction has been made for
Alexandrov space with bilaterally bounded curvature by Igor Nikolaev (see [N]).¶Naturally, as we have a more general situation,
the constructed transportation does not have such good properties as in the case of bilaterally bounded curvature. In particular,
we cannot prove the uniqueness in any good sense. Nevertheless the constructed transportation is enough for the most important
applications such as Synge's lemma and Frankel's theorem. Recently by using this parallel transportation together with techniques
of harmonic functions on Alexandrov space, we have proved an isoperimetric inequality of Gromov's type.¶Author is indebted
to Stephanie Alexander, Yuri Burago and Grisha Perelman for their willingness to understand, interest and important remarks.
Ricci curvature bounds in Riemannian geometry are known to be equivalent to the weak convexity (convexity along at least one
geodesic between any two points) of certain functionals in the space of probability measures. We prove that the weak convexity
can be reinforced into strong (usual) convexity, thus solving a question left open in Lott and Villani (Ann of Math, to appear).
We generalize the Alexandrov–Toponogov comparison theorems to Finsler manifolds. Under suitable upper (lower, resp.) bounds
on the flag and tangent curvatures together with the 2-uniform convexity (smoothness, resp.) of tangent spaces, we show the
2-uniform convexity (smoothness, resp.) of Finsler manifolds. As applications, we prove the almost everywhere existence of
the second order differentials of semi-convex functions and of c-concave functions with the quadratic cost function.
We give a necessary and sufficient condition on a Randers space for the existence of a measure for which Shenʼs S-curvature vanishes everywhere. Moreover, if it exists, such a measure coincides with the Busemann–Hausdorff measure up to a constant multiplication.
The topic of this paper are convexity properties of free energy functionals on the space P2(M) of probability measures over a Riemannian manifold. As applications, we obtain contraction properties of nonlinear diffusions on Rn or on a Riemannian manifold M, regarding them as gradient flows of appropriate free energy functionals. In particular, we present extensions of the Bakry–Emery criterion to nonlinear equations.RésuméCet article traite de propriétés de convexité de fonctionnelles d'énergie libre sur l'espace P2(M) des mesures de probabilités sur une variété riemannienne. Comme applications nous obtenons des propriétés de contraction de diffusions non-linéaires, sur Rn ou sur une variété riemannienne M, en les considérant comme des flots de gradients pour des fonctionnelles d'énergie libre apropriées. En particulier, nous présentons des extensions du critère de Bakry–Emery à des équations non-linéaires.
We show that transport inequalities, similar to the one derived by M. Talagrand (1996, Geom. Funct. Anal.6, 587–600) for the Gaussian measure, are implied by logarithmic Sobolev inequalities. Conversely, Talagrand's inequality implies a logarithmic Sobolev inequality if the density of the measure is approximately log-concave, in a precise sense. All constants are independent of the dimension and optimal in certain cases. The proofs are based on partial differential equations and an interpolation inequality involving the Wasserstein distance, the entropy functional, and the Fisher information.
The purpose of this paper is to show that in a finite dimensional metric space with Alexandrov's curvature bounded below, Monge's transport problem for the quadratic cost admits a unique solution.
We give sufficient conditions for a measured length space (X,d,ν) to admit local and global Poincaré inequalities, along with a Sobolev inequality. We first introduce a condition DM on (X,d,ν), defined in terms of transport of measures. We show that DM, together with a doubling condition on ν, implies a scale-invariant local Poincaré inequality. We show that if (X,d,ν) has nonnegative N-Ricci curvature and has unique minimizing geodesics between almost all pairs of points then it satisfies DM, with constant N2. The condition DM is preserved by measured Gromov–Hausdorff limits. We then prove a Sobolev inequality for measured length spaces with N-Ricci curvature bounded below by K>0. Finally we derive a sharp global Poincaré inequality.
We define the coarse Ricci curvature of metric spaces in terms of how much small balls are closer (in Wasserstein transportation distance) than their centers are. This definition naturally extends to any Markov chain on a metric space. For a Riemannian manifold this gives back, after scaling, the value of Ricci curvature of a tangent vector. Examples of positively curved spaces for this definition include the discrete cube and discrete versions of the Ornstein–Uhlenbeck process. Moreover this generalization is consistent with the Bakry–Émery Ricci curvature for Brownian motion with a drift on a Riemannian manifold.Positive Ricci curvature is shown to imply a spectral gap, a Lévy–Gromov–like Gaussian concentration theorem and a kind of modified logarithmic Sobolev inequality. The bounds obtained are sharp in a variety of examples.
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.
310 ZHONGMIN Page 6. both arguments in [C1] and [GP] were carried out using Toponogov's comparison theorem. However, we can show that, in a manifold