##
**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].

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].

Reviewer: Sergei V. Buyalo (St. Petersburg)

### MSC:

53C23 | Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces |

### Keywords:

metric measure spaces; optimal mass transport; relative entropy of a measure; lower Ricci curvature bounds### Citations:

Zbl 1105.53035
Full Text:
DOI

### References:

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