## Acta Mathematica

### On the geometry of metric measure spaces

Karl-Theodor Sturm

#### Abstract

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 L2-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 $\xi \in TM$.

The crucial point is that our lower curvature bounds are stable under an appropriate notion of D-convergence of metric measure spaces. We define a complete and separable length metric D on the family of all isomorphism classes of normalized metric measure spaces. The metric D has a natural interpretation, based on the concept of optimal mass transportation.

We also prove that the family of normalized metric measure spaces with doubling constant ⩽ C is closed under D-convergence. Moreover, the family of normalized metric measure spaces with doubling constant ⩽ C and diameter ⩽ L is compact under D-convergence.

#### Article information

Source
Acta Math., Volume 196, Number 1 (2006), 65-131.

Dates
Revised: 11 January 2006
First available in Project Euclid: 31 January 2017

https://projecteuclid.org/euclid.acta/1485891805

Digital Object Identifier
doi:10.1007/s11511-006-0002-8

Mathematical Reviews number (MathSciNet)
MR2237206

Zentralblatt MATH identifier
1106.53032

Rights

#### Citation

Sturm, Karl-Theodor. On the geometry of metric measure spaces. Acta Math. 196 (2006), no. 1, 65--131. doi:10.1007/s11511-006-0002-8. https://projecteuclid.org/euclid.acta/1485891805

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