Abstract
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 Poincaré inequalities on balls. In particular, it allows to construct canonical Dirichlet forms with Gaussian upper and lower bounds for the corresponding heat kernels.
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References
Ambrosio, L., Tilli, P.: Topics on Analysis in Metric Spaces. Oxford Lecture Series in Mathematics and its Applications, 25. Oxford University Press, Oxford (2004)
Bakry, D., Émery, M.: Diffusions hypercontractives. In: Séminaire de probabilités, XIX, 1983/84. Lecture Notes in Math., 1123, pp. 177–206. Springer, Berlin (1985)
Bakry, D., Qian, Z.: Some new results on eigenvectors via dimension, diameter, and Ricci curvature. Adv. Math. 155, 98–153 (2000)
Chavel, I.: Riemannian Geometry—a Modern Introduction. Cambridge Tracts in Mathematics, 108. Cambridge University Press, Cambridge (1993)
Cheeger, J.: Differentiability of Lipschitz functions on metric measure spaces. Geom. Funct. Anal. 9, 428–517 (1999)
Cordero-Erausquin, D., McCann, R.J., Schmuckenschläger, M.: A Riemannian interpolation inequality à la Borell, Brascamp and Lieb. Invent. Math. 146, 219–257 (2001)
Gromov, M.: Metric Structures for Riemannian and non-Riemannian Spaces. Progress in Mathematics, 152. Birkhäuser Boston, Boston, MA (1999)
Hajłasz, P.: Sobolev spaces on an arbitrary metric space. Potential Anal. 5, 403–415 (1996)
Heinonen, J.: Lectures on Analysis on Metric Spaces. Springer, New York (2001)
Korevaar, N.J., Schoen, R.M.: Sobolev spaces and harmonic maps for metric space targets. Comm. Anal. Geom. 1, 561–659 (1993)
Kuwae, K., Shioya, T.: On generalized measure contraction property and energy functionals over Lipschitz maps. Potential Anal. 15, 105–121 (2001)
Kuwae, K., Shioya, T.: Sobolev and Dirichlet spaces over maps between metric spaces. J. Reine Angew. Math. 555, 39–75 (2003)
Lott, J., Villani, C.: Ricci curvature for metric-measure spaces via optimal transport. Preprint (2005)
Lott J., Villani, C.: Weak curvature conditions and Poincaré inequalities. Preprint (2005)
Ohta, S.-I.: On measure contraction property of metric measure spaces. Preprint (2006)
von Renesse, M.-K.: Local Poincaré via transportation. To appear in Math. Z
Sturm, K.-T.: Analysis on local Dirichlet spaces. III. The parabolic Harnack inequality. J. Math. Pures Appl. 75, 273–297 (1996)
Sturm, K.-T.: Diffusion processes and heat kernels on metric spaces. Ann. Probab. 26, 1–55 (1998)
Sturm, K.-T.: Convex functionals of probability measures and nonlinear diffusions on manifolds. J. Math. Pures Appl. 84, 149–168 (2005)
Sturm, K.-T.: A curvature-dimension condition for metric measure spaces. C. R. Math. Acad. Sci. Paris 342, 197–200 (2006)
Sturm, K.-T.: On the geometry of metric measure spaces. I. Acta Math. 196, 65–131 (2006)
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Sturm, KT. On the geometry of metric measure spaces. II. Acta Math 196, 133–177 (2006). https://doi.org/10.1007/s11511-006-0003-7
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DOI: https://doi.org/10.1007/s11511-006-0003-7