On the geometry of metric measure spaces. II

Karl-Theodor Sturm Institut für Angewandte Mathematik, Universität Bonn

TBD mathscidoc:1701.331970

Acta Mathematica, 196, (1), 133-177, 2005.3
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($M$) ⩽$N$.
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  title={On the geometry of metric measure spaces. II},
  author={Karl-Theodor Sturm},
  booktitle={Acta Mathematica},
Karl-Theodor Sturm. On the geometry of metric measure spaces. II. 2005. Vol. 196. In Acta Mathematica. pp.133-177. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203348930443679.
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