Acta Mathematica

, Volume 196, Issue 1, pp 133–177 | Cite as

On the geometry of metric measure spaces. II

  • Karl-Theodor Sturm


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.


Riemannian Manifold Measure Space Ricci Curvature Dirichlet Form Minkowski Inequality 
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© Springer-Verlag 2006

Authors and Affiliations

  1. 1.Institut für Angewandte MathematikUniversität BonnBonnGermany

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