## On the geometry of metric measure spaces. II.(English)Zbl 1106.53032

This is the second of two papers on the geometry of metric measure spaces $$(M,d,m)$$ presenting a concept of lower Ricci curvature bounds. In this paper, metric measure spaces satisfying a curvature-dimension condition $$\text{CD}(K,N)$$, where $$K\in{\mathbb R}$$, $$N\geq 1$$, are studied. The condition $$\text{CD}(K,N)$$ is more restrictive than the previous condition $$\underline{\text{Curv}}(M,d,m)\geq K$$ studied in the first paper [[S]: K.-T. Sturm, Acta Math. 196, No. 1, 65–131 (2006; Zbl 1105.53035)]. The additional parameter $$N$$ plays the role of an upper bound for the dimension, and loosely speaking the dimension-independent condition $$\underline{\text{Curv}}(M,d,m)\geq K$$ is the borderline case of $$\text{CD}(K,N)$$ for $$N=\infty$$.
As in [S], the definition of the curvature-dimension condition is based on a convexity property for suitable functionals on the space $${\mathcal P}_2(M)$$ of probability measures on $$M$$. For finite $$N$$, the basic object is the Rényi entropy functional $S_N(\rho m| m)=-\int\rho^{1-1/N}\,dm.$ A metric measure space $$(M,d,m)$$ satisfies the curvature-dimension condition $$\text{CD}(0,N)$$ for some $$N\geq 1$$ if and only if the Rényi entropy functionals $$S_{N'}(\cdot\,| m)$$ for all $$N'\geq N$$ are weakly convex on the space $${\mathcal P}_2(M,d,m)$$ of w.r.t $$m$$ absolutely continuous probability measures on $$M$$: for each pair $$\nu_0$$, $$\nu_1\in{\mathcal P}_2(M,d,m)$$ there exists a geodesic $$\Gamma:[0,1]\to{\mathcal P}_2(M,d,m)$$ connecting $$\nu_0$$ and $$\nu_1$$, with $S_{N'}(\Gamma(t)| m)\leq(1-t)S_{N'}(\nu_0| m)+ tS_{N'}(\nu_1| m)$ for all $$t\in[0,1]$$ and $$N'\geq N$$ (in the case $$K\neq 0$$, the definition is more involved).
The main results of this paper are:
For Riemannian manifolds, $$\text{CD}(K,N)$$ is equivalent to the condition $$\text{Ric}_M(\xi,\xi)\geq K| \xi| ^2$$ for all $$\xi\in TM$$ and $$\dim M\leq N$$. Furthermore, for each metric measure space $$(M,d,m)$$ satisfying $$\text{CD}(K,N)$$, the support of $$M$$ has Hausdorff dimension $$\leq N$$.
The curvature-dimension condition $$\text{CD}(K,N)$$ is stable under $${\mathbf D}$$-convergence (the metric $${\mathbf D}$$ was introduced in [S]), in particular, $$\text{CD}(K,N)$$ is stable under measured Gromov-Hausdorff convergence. Moreover, for any reals $$K$$, $$N\geq 1$$, $$L\geq 0$$ the family of normalized metric measure spaces with condition $$\text{CD}(K,N)$$ and diameter $$\leq L$$ is $${\mathbf D}$$-compact (compact w.r.t. measured Gromov-Hausdorff topology).
Condition $$\text{CD}(K,N)$$ implies a generalized version of the Brunn-Minkowski inequality, e.g., if $$K=0$$ then $m(A_t)^{1/N}\geq(1-t)m(A_0)^{1/N}+tm(A_1)^{1/N}$ for each $$t\in[0,1]$$ and any pair of sets $$A_0$$, $$A_1\subset M$$, where $$A_t$$ denotes the set of all possible points $$\gamma_t$$ on geodesics in $$M$$ with endpoints $$\gamma_0\in A_0$$, $$\gamma_1\in A_1$$;
Condition $$\text{CD}(K,N)$$ implies the Bishop-Gromov volume comparison theorem, e.g., if $$K=0$$ then ${m(B_r(x))\over m(B_R(x))}\geq\left({r\over R}\right)^N.$ Condition $$\text{CD}(K,N)$$ for some $$K>0$$ provides a sharp upper bound on the diameter (Bonnet-Myers theorem): $L\leq\pi\sqrt{{N-1\over k}}.$ Under minimal regularity assumptions, condition $$\text{CD}(K,N)$$ implies property $$\text{MCP}(K,N)$$ (measure contraction property): roughly spoken, $$\text{CD}(K,N)$$ is a condition on the optimal transport between any pair of (absolutely continuous) probability measures on $$M$$, whereas $$\text{MCP}(K,N)$$ is a condition on the optimal transport between Dirac measures and the uniform distribution on $$M$$. Most of the above results also remain true with condition $$\text{MCP}(K,N)$$ in place of condition $$\text{CD}(K,N)$$. Furthermore, every (complete locally compact) Alexandrov space with curvature $$\geq\kappa$$ and with finite Hausdorff dimension $$n$$ satisfies property $$\text{MCP}((n-1)\kappa,n)$$.
Of particular interest are the analytic consequences of $$\text{MCP}(K,N)$$. It allows one to construct a canonical Dirichlet form and a canonical Laplace operator on $$L_2(M,m)$$, it implies a local Poincaré inequality, a scale invariant Harnack inequality, and Gaussian estimates for heat kernel, and it yields Hölder continuity of harmonic functions.
Similar results are independently obtained in 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

Zbl 1105.53035
Full Text:

### References:

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