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On the geometry of metric measure spaces. I. (English) Zbl 1105.53035
This is the first of two papers on the geometry of metric measure spaces presenting a concept of lower Ricci curvature bounds. In contrast to lower sectional curvature bounds, which can be defined in terms of the metric structure of a general metric space, a reference measure $$m$$ has to be specified to define lower Ricci curvature bounds. The basic idea of the approach can loosely be described as follows. Given a metric space $$(M,d)$$, we consider the space $${\mathcal P}_2(M)$$ of all probability measures on $$M$$ with the distance $d_W(\mu,\nu)=\inf\left\{\left( \int_{M\times M}d^2(x,y)dq(x,y)\right)^{1/2}: q\;{\text{is a coupling of}}\;\mu\;\text{and}\;\nu \right\},$ that is, $$d_W(\mu,\nu)$$ is the cost of an optimal mass transport between the measures $$\mu$$ and $$\nu$$ with the cost function $$d^2(x,y)$$. Now, fixing a reference measure $$m$$ on $$M$$, we say that the metric measure space $$(M,d,m)$$ has (Ricci) curvature $$\geq K$$ for some $$K\in\mathbb{R}$$ if and only if a relative entropy $$\nu\mapsto \text{ Ent}(\nu| m)$$ is a $$K$$-convex function on $${\mathcal P}_2(M)$$.
In this first paper, a dimension-independent concept of lower Ricci curvature bounds is presented, that is, the relative entropy for $$\nu=\rho m$$ is defined by $\text{ Ent}(\nu| \mu)=\lim_{\varepsilon\searrow 0} \int_{\rho>\varepsilon}\rho\log\rho\, dm.$ The main results of this paper are:
A complete and separable metric D on the family of all isomorphism classes of normalized metric measure spaces is defined via optimal mass transport. It is a length metric. D-convergence is weaker than measured Gromov-Hausdorff convergence. Both are equivalent on each family of compact metric measure spaces with full support and uniform bounds for the doubling constant and the diameter. Each of these families is D-compact;
A notion of lower curvature bound $$\underline{\text{Curv}}(M,d,m)$$ for a metric measure space $$(M,d,m)$$, based on a convexity property of the relative entropy $$\text{Ent}(\cdot\,| m)$$ with respect to the reference measure $$m$$, is introduced. For Riemannian manifolds, $$\underline{\text{Curv}}(M,d,m)\geq K$$ if and only if the Ricci curvature satisfies $$\text{ Ric}_M(\xi,\xi)\geq K| \xi| ^2$$ for all $$\xi\in TM$$. Local lower curvature bounds imply global lower curvature bounds.
Lower curvature bounds are stable under D-convergence, in particular, lower curvature bounds are stable under measured Gromov-Hausdorff convergence. Lower curvature bounds of the form $$\underline{\text{Curv}}(M,d,m)\geq K$$ imply (sharp) estimates for the volume growth of concentric balls in terms of squared exponentially growing functions. These estimates are best possible in general because the concept of lower curvature bounds is dimension-independent. (The second paper treats the finite-dimensional case, where metric measure spaces with more restrictive lower curvature bounds depending on dimension are considered. For such spaces the volume growth of concentric balls holds in precise form of the Bishop-Gromov volume comparison).
Similar results are independently obtained by [J. Lott and C. Villani [Ricci curvature for metric-measure spaces via optimal mass transport, arXive:math.DG/0412127 (to appear in Ann. Math.)] see also a survey by J. Lott [Optimal transport and Ricci curvature for metric-measure spaces, arXive:math.DG/0610154].

##### MSC:
 53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces 28D20 Entropy and other invariants
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