On the geometry of metric measure spaces

Karl-Theodor Sturm Institut für Angewandte Mathematik, Universität Bonn

TBD mathscidoc:1701.331969

Acta Mathematica, 196, (1), 65-131, 2004.10
We introduce and analyze lower ($Ricci$) curvature bounds $ \underline{{Curv}} {\left( {M,d,m} \right)} $ ⩾$K$for metric measure spaces $ {\left( {M,d,m} \right)} $ . Our definition is based on convexity properties of the relative entropy $ Ent{\left( { \cdot \left| m \right.} \right)} $ regarded as a function on the$L$_{2}-Wasserstein space of probability measures on the metric space $ {\left( {M,d} \right)} $ . Among others, we show that $ \underline{{Curv}} {\left( {M,d,m} \right)} $ ⩾$K$implies estimates for the volume growth of concentric balls. For Riemannian manifolds, $ \underline{{Curv}} {\left( {M,d,m} \right)} $ ⩾$K$if and only if $ Ric_{M} {\left( {\xi ,\xi } \right)} $ ⩾$K$ $ {\left| \xi \right|}^{2} $ for all $ \xi \in TM $ .
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@inproceedings{karl-theodor2004on,
  title={On the geometry of metric measure spaces},
  author={Karl-Theodor Sturm},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203348822929678},
  booktitle={Acta Mathematica},
  volume={196},
  number={1},
  pages={65-131},
  year={2004},
}
Karl-Theodor Sturm. On the geometry of metric measure spaces. 2004. Vol. 196. In Acta Mathematica. pp.65-131. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203348822929678.
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