On the geometry of metric measure spaces. II
On the geometry of metric measure spaces. II
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DOI：
10.1007/s11511-006-0003-7
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发表时间：
2006-01-01
期刊：
影响因子：
3.7
通讯作者：
Sturm, Karl-Theodor
中科院分区：
文献类型：
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作者：
Sturm, Karl-Theodor
我们引入度量度量空间的曲率-维条件CD(K，N)。它比[Sturm K-T(2006)]中引入的曲率界对度量空间的几何有更多的限制。I.数学学报196：65-131])，其被恢复为边界案例CD(K， ∞)。附加的实参数N起到了维度的广义上界的作用。对于黎曼流形，CD(K，N)等价于Dim(M) ⩽n，曲率-维条件CD(K，N)在收敛条件下是稳定的。对于实数的任意三元组K，N，l，具有Cd(K，N)和直径 Lis紧的正规化度量空间族(M，⩽d，m)。条件Cd(K，N)意味着Brunn-Minkowski不等式、Bishop-Gromov体积比较定理和Bonnet-Myers定理的尖锐版本。此外，它还蕴含了球上的倍增性质和局部尺度不变的Poincaré不等式。特别地，它允许构造具有相应热核的高斯上界和下界的规范Dirichlet形式。
We introduce a curvature-dimension condition CD (K,N) for metric measure spaces. It is more restrictive than the curvature bound(introduced in [Sturm K-T (2006) On the geometry of metric measure spaces. I. Acta Math 196:65–131]) which is recovered as the borderline case CD(K, ∞). The additional real parameterNplays the role of a generalized upper bound for the dimension. For Riemannian manifolds, CD(K,N) is equivalent toand dim(M) ⩽N.The curvature-dimension condition CD(K,N) is stable under convergence. For any triple of real numbersK,N,Lthe family of normalized metric measure spaces (M, d,m) with CD(K,N) and diameter ⩽Lis compact.Condition CD(K,N) implies sharp version of the Brunn–Minkowski inequality, of the Bishop–Gromov volume comparison theorem and of the Bonnet–Myers theorem. Moreover, it implies the doubling property and local, scale-invariant Poincaré inequalities on balls. In particular, it allows to construct canonical Dirichlet forms with Gaussian upper and lower bounds for the corresponding heat kernels.