On the geometry of metric measure spaces. I. (English) Zbl 1105.53035

This is the first of two papers on the geometry of metric measure spaces presenting a concept of lower Ricci curvature bounds. In contrast to lower sectional curvature bounds, which can be defined in terms of the metric structure of a general metric space, a reference measure \(m\) has to be specified to define lower Ricci curvature bounds. The basic idea of the approach can loosely be described as follows. Given a metric space \((M,d)\), we consider the space \({\mathcal P}_2(M)\) of all probability measures on \(M\) with the distance \[ d_W(\mu,\nu)=\inf\left\{\left( \int_{M\times M}d^2(x,y)dq(x,y)\right)^{1/2}: q\;{\text{is a coupling of}}\;\mu\;\text{and}\;\nu \right\}, \] that is, \(d_W(\mu,\nu)\) is the cost of an optimal mass transport between the measures \(\mu\) and \(\nu\) with the cost function \(d^2(x,y)\). Now, fixing a reference measure \(m\) on \(M\), we say that the metric measure space \((M,d,m)\) has (Ricci) curvature \(\geq K\) for some \(K\in\mathbb{R}\) if and only if a relative entropy \(\nu\mapsto \text{ Ent}(\nu| m)\) is a \(K\)-convex function on \({\mathcal P}_2(M)\).
In this first paper, a dimension-independent concept of lower Ricci curvature bounds is presented, that is, the relative entropy for \(\nu=\rho m\) is defined by \[ \text{ Ent}(\nu| \mu)=\lim_{\varepsilon\searrow 0} \int_{\rho>\varepsilon}\rho\log\rho\, dm. \] The main results of this paper are:
A complete and separable metric D on the family of all isomorphism classes of normalized metric measure spaces is defined via optimal mass transport. It is a length metric. D-convergence is weaker than measured Gromov-Hausdorff convergence. Both are equivalent on each family of compact metric measure spaces with full support and uniform bounds for the doubling constant and the diameter. Each of these families is D-compact;
A notion of lower curvature bound \(\underline{\text{Curv}}(M,d,m)\) for a metric measure space \((M,d,m)\), based on a convexity property of the relative entropy \(\text{Ent}(\cdot\,| m)\) with respect to the reference measure \(m\), is introduced. For Riemannian manifolds, \(\underline{\text{Curv}}(M,d,m)\geq K\) if and only if the Ricci curvature satisfies \(\text{ Ric}_M(\xi,\xi)\geq K| \xi| ^2\) for all \(\xi\in TM\). Local lower curvature bounds imply global lower curvature bounds.
Lower curvature bounds are stable under D-convergence, in particular, lower curvature bounds are stable under measured Gromov-Hausdorff convergence. Lower curvature bounds of the form \(\underline{\text{Curv}}(M,d,m)\geq K\) imply (sharp) estimates for the volume growth of concentric balls in terms of squared exponentially growing functions. These estimates are best possible in general because the concept of lower curvature bounds is dimension-independent. (The second paper treats the finite-dimensional case, where metric measure spaces with more restrictive lower curvature bounds depending on dimension are considered. For such spaces the volume growth of concentric balls holds in precise form of the Bishop-Gromov volume comparison).
Similar results are independently obtained by [J. Lott and C. Villani [Ricci curvature for metric-measure spaces via optimal mass transport, arXive:math.DG/0412127 (to appear in Ann. Math.)] see also a survey by J. Lott [Optimal transport and Ricci curvature for metric-measure spaces, arXive:math.DG/0610154].


53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
28D20 Entropy and other invariants
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[1] Alexandrov, A.D.: A theorem on triangles in a metric space and some applications. Trudy Mat. Inst. Steklov 38, 5–23 (1951) (Russian; translated into German and com-bined with more material in [2])
[2] Alexandrov, A.D.: Über eine Verallgemeinerung der Riemannschen Geometrie. Schr. Forschungsinst. Math. Berlin 1, 33–84 (1957)
[3] Ambrosio, L., Tilli, P.: Topics on Analysis in Metric Spaces. Oxford Lecture Series in Mathematics and its Applications, 25. Oxford University Press, Oxford (2004) · Zbl 1080.28001
[4] Bakry, D., Émery, M.: Diffusions hypercontractives. In: Séminaire de Probabilités, XIX, 1983/84. Lecture Notes in Math., 1123, pp. 177–206. Springer, Berlin (1985)
[5] Bobkov, S.G., Götze, F.: Exponential integrability and transportation cost related to logarithmic Sobolev inequalities. J. Funct. Anal. 163, 1–28 (1999) · Zbl 0924.46027
[6] Brenier, Y.: Polar factorization and monotone rearrangement of vector-valued functions. Comm. Pure Appl. Math. 44, 375–417 (1991) · Zbl 0738.46011
[7] Bridson, M.R., Haefliger, A.: Metric Spaces of Non-positive Curvature. Grundlehren der Mathematischen Wissenschaften, 319. Springer, Berlin (1999) · Zbl 0988.53001
[8] Burago, D., Burago, Y., Ivanov, S.: A Course in Metric Geometry. Graduate Studies in Mathematics, 33. Amer. Math. Soc., Providence, RI (2001) · Zbl 0981.51016
[9] Burago, Y., Gromov, M., Perelman, G.: A. D. Aleksandrov spaces with curvatures bounded below. Uspekhi Mat. Nauk 47(2), 3–51 (1992), 222 (Russian); English translation in Russian Math. Surveys 47(2), 1–58 (1992)
[10] Chavel, I.: Riemannian Geometry–a Modern Introduction. Cambridge Tracts in Mathematics, 108. Cambridge University Press, Cambridge (1993) · Zbl 0810.53001
[11] Cheeger, J.: Differentiability of Lipschitz functions on metric measure spaces. Geom.Funct. Anal. 9, 428–517 (1999) · Zbl 0942.58018
[12] Cheeger, J., Colding, T.H.: On the structure of spaces with Ricci curvature bounded below I, II, III. J. Differential Geom. 46, 406–480 (1997); Ibid, 54, 13–35 (2000); Ibid, 54, 37–74 (2000) · Zbl 0902.53034
[13] Cordero-Erausquin, D., McCann, R.J., Schmuckenschläger, M.: A Riemannian interpolation inequality à la Borell, Brascamp and Lieb. Invent. Math. 146, 219–257 (2001) · Zbl 1026.58018
[14] Davies, E.B.: Heat Kernels and Spectral Theory. Cambridge Tracts in Mathematics, 92. Cambridge University Press, Cambridge (1989) · Zbl 0699.35006
[15] Del Pino, M., Dolbeault, J.: Best constants for Gagliardo–Nirenberg inequalities and applications to nonlinear diffusions. J. Math. Pures Appl. 81, 847–875 (2002) · Zbl 1112.35310
[16] Dudley, R. M.: Real Analysis and Probability. Wadsworth & Brooks/Cole, Pacific Grove, CA (1989) · Zbl 0686.60001
[17] Feyel, D., Üstünel, A.S.: Monge–Kantorovitch measure transportation and Monge–Ampère equation on Wiener space. Probab. Theory Related Fields 128, 347–385 (2004) · Zbl 1055.60052
[18] Feyel D., Üstünel, A.S.: The strong solution of the Monge–Ampère equation on the Wiener space for log-concave densities. C. R. Math. Acad. Sci. Paris, 339, 49–53 (2004) · Zbl 1051.60070
[19] Fukaya, K.: Collapsing of Riemannian manifolds and eigenvalues of Laplace operator. Invent. Math. 87, 517–547 (1987) · Zbl 0589.58034
[20] Gromov, M.: Groups of polynomial growth and expanding maps. Inst. Hautes Études Sci. Publ. Math. 53, 53–73 (1981) · Zbl 0474.20018
[21] Gromov, M.: Structures Mètriques pour les Variètès Riemanniennes. Textes Mathèmatiques, 1 · Zbl 0509.53034
[22] Gromov, M.: Metric Structures for Riemannian and non-Riemannian Spaces. Progress in Mathematics, 152. Birkhäuser Boston, Boston, MA, Based on [21] (1999) · Zbl 0953.53002
[23] Grove, K., Petersen, P.: Manifolds near the boundary of existence. J. Differential Geom. 33, 379–394 (1991) · Zbl 0729.53045
[24] Hajłasz, P., Koskela, P.: Sobolev meets Poincarè. C. R. Acad. Sci. Paris Sèr. I Math. 320, 1211–1215 (1995) · Zbl 0837.46024
[25] Hajłasz, P., Koskela, P.: Sobolev met Poincaré. Mem. Amer. Math. Soc. 145 (2000) · Zbl 0954.46022
[26] Heinonen, J.: Lectures on Analysis on Metric Spaces. Universitext. Springer, New York (2001) · Zbl 0985.46008
[27] Jordan, R., Kinderlehrer, D., Otto, F.: The variational formulation of the Fokker–Planck equation. SIAM J. Math. Anal. 29, 1–17 (1998) · Zbl 0915.35120
[28] Kantorovich, L.V.: On the translocation of masses. C. R. (Doklady) Acad. Sci. URSS 37, 199–201 (1942) · Zbl 0061.09705
[29] Kantorovich, L.V., Rubinshteĭn, G.S.: On a functional space and certain extremum problems. Dokl. Akad. Nauk SSSR 115, 1058–1061 (1957) · Zbl 0081.11501
[30] Kasue, A.: Convergence of Riemannian manifolds and Laplace operators II. Preprint (2004) · Zbl 0806.53048
[31] Kasue, A., Kumura, H.: Spectral convergence of Riemannian manifolds. Tohoku Math. J. 46, 147–179 (1994) · Zbl 0814.53035
[32] Kigami, J.: Analysis on Fractals. Cambridge Tracts in Mathematics, 143. Cambridge Uni-versity Press, Cambridge (2001) · Zbl 0998.28004
[33] Koskela, P.: Upper gradients and Poincaré inequalities. In: Lecture Notes on Analysis in Metric Spaces (Trento, 1999), pp. 55–69. Appunti Corsi Tenuti Docenti Sc. Scuola Norm. Sup., Pisa (2000)
[34] Kuwae, K., Shioya, T.: Convergence of spectral structures: a functional analytic theory and its applications to spectral geometry. Comm. Anal. Geom. 11, 599–673 (2003) · Zbl 1092.53026
[35] Ledoux, M.: The Concentration of Measure Phenomenon. Mathematical Surveys and Monographs, 89. Amer. Math. Soc., Providence, RI (2001)
[36] Li, P., Yau, S.-T.: On the parabolic kernel of the Schrödinger operator. Acta Math. 156, 153–201 (1986) · Zbl 0611.58045
[37] Lott, J., Villani, C.: Ricci curvature for metric-measure spaces via optimal transport. Preprint (2005) · Zbl 1178.53038
[38] McCann, R.J.: Existence and uniqueness of monotone measure-preserving maps. Duke Math. J. 80, 309–323 (1995) · Zbl 0873.28009
[39] McCann, R.J.: A convexity principle for interacting gases. Adv. Math. 128, 153–179 (1997) · Zbl 0901.49012
[40] Monge, G.: Mémoire sur la théorie des déblais et des remblais. In: Histoire de l’Académie Royale des Sciences. Paris (1781)
[41] Otto, F.: The geometry of dissipative evolution equations: the porous medium equation. Comm. Partial Differential Equations 26, 101–174 (2001) · Zbl 0984.35089
[42] Otto, F., Villani, C.: Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality. J. Funct. Anal. 173, 361–400 (2000) · Zbl 0985.58019
[43] Perelman, G.: The entropy formula for the Ricci flow and its geometric applications. Preprint (2002) · Zbl 1130.53001
[44] Plaut, C.: Metric spaces of curvature . In: Handbook of Geometric Topology, pp. 819–898. North-Holland, Amsterdam (2002) · Zbl 1011.57002
[45] Rachev, S.T., Rüschendorf, L.: Mass Transportation Problems. Vol. I. Probability and its Applications (New York). Springer, New York (1998) · Zbl 0990.60500
[46] von Renesse, M.-K., Sturm, K.-T.: Transport inequalities, gradient estimates, entropy, and Ricci curvature. Comm. Pure Appl. Math. 58, 923–940 (2005) · Zbl 1078.53028
[47] Saloff-Coste, L.: Aspects of Sobolev-type Inequalities. London Math. Soc. Lecture Note Series, 289. Cambridge University Press, Cambridge (2002) · Zbl 0991.35002
[48] Sturm, K.-T.: Diffusion processes and heat kernels on metric spaces. Ann. Probab. 26, 1–55 (1998) · Zbl 0936.60074
[49] Sturm, K.-T.: Metric spaces of lower bounded curvature. Exposition. Math. 17, 35–47 (1999) · Zbl 0983.53025
[50] Sturm, K.-T.: Probability measures on metric spaces of nonpositive curvature. In: Heat Kernels and Analysis on Manifolds, Graphs, and Metric Spaces (Paris, 2002), pp. 357–390. Contemp. Math., 338. Amer. Math. Soc., Providence, RI (2003)
[51] Sturm, K.-T.: Convex functionals of probability measures and nonlinear diffusions on manifolds. J. Math. Pures Appl. 84, 149–168 (2005) · Zbl 1259.49074
[52] Sturm, K.-T.: Generalized Ricci bounds and convergence of metric measure spaces. C. R. Math. Acad. Sci. Paris 340, 235–238 (2005) · Zbl 1092.28010
[53] Sturm, K.-T.: On the geometry of metric measure spaces. II. Acta Math. 196, 133–177 (2006) · Zbl 1106.53032
[54] Talagrand, M.: Concentration of measure and isoperimetric inequalities in product spaces. Inst. Hautes Études Sci. Publ. Math., 73–205 (1995) · Zbl 0864.60013
[55] Villani, C.: Topics in Optimal Transportation. Graduate Studies in Mathematics, 58. Amer. Math. Soc., Providence, RI (2003) · Zbl 1106.90001
[56] Wasserstein [Vasershtein], L.N.: Markov processes over denumerable products of spaces describing large system of automata. Problemy Peredači Informacii 5(3), 64–72 (1969) (Russian). English translation in Problems of Information Transmission 5(3), 47–52 (1969)
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