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2006 On the geometry of metric measure spaces. II
Karl-Theodor Sturm
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Acta Math. 196(1): 133-177 (2006). DOI: 10.1007/s11511-006-0003-7


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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Karl-Theodor Sturm. "On the geometry of metric measure spaces. II." Acta Math. 196 (1) 133 - 177, 2006.


Received: 18 March 2005; Revised: 10 June 2005; Published: 2006
First available in Project Euclid: 31 January 2017

zbMATH: 1106.53032
MathSciNet: MR2237207
Digital Object Identifier: 10.1007/s11511-006-0003-7

Rights: 2006 © Springer-Verlag

Vol.196 • No. 1 • 2006
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