On the geometry of metric measure spaces. II

By Karl‐Theodor Sturm. 2006

1,226 citations recorded by the source corpus.

AbstractWe introduce a curvature-dimension condition CD (K, N) for metric measure spaces. It is more restrictive than the curvature bound $\underline{{{\text{Curv}}}} {\left( {M,{\text{d}},m} \right)} > K$ (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 parameter N plays the role of a generalized upper bound for the dimension. For Riemannian manifolds, CD(K, N) is equivalent to $${\text{Ric}}_{M} {\left( {\xi ,\xi } \right)} > K{\left| \xi \right|}^{2} $$ and dim(M) ⩽ N.The curvature-dimension condition CD(K, N) is stable under convergence. For any triple of real numbers K, N, L the family of normalized metric measure spaces (M, d, m) with CD(K, N) and diameter ⩽ L is 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.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/84357afeddd5c45c4571712d578cc6fe37093979

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Citations in the corpus

Sharp and rigid isoperimetric inequalities in metric-measure spaces with lower Ricci curvature bounds (2015, background): “d dimension bounded above by N” for m.m.s.. 1.2. Ricci curvature lower bounds for metric measure spaces. The investigation about the topic began with the seminal papers of Lott-Villani [45] and Sturm [64, 65], though has been adapted considerably since the work of Bacher-Sturm [9] and Ambrosio-Gigli”
Optimal transport and Perelman’s reduced volume (2008, background): “In another direction, there has been recent work showing the equivalence between the nonnegative Ricci curvature of a Riemannian manifold M , and the convexity (in time) of certain entropy functions in an optimal transport problem on M [4, 15, 19, 23, 24, 25].”
Cheeger-harmonic functions in metric measure spaces revisited (2013, background): “On a complete metric space satisfying Lott-Sturm-Villani’ s curvature conditionCD(K,N) for someK ∈ R andN ∈ (1,∞) (see [24, 33], in [24] onlyCD(0,N) condition is introduced), Sturm [33, Corollary 2.”
Transportation-information inequalities for Markov processes (2007, methodology): “Sturm.”
Convergence in distribution of random metric measure spaces (Λ-coalescent measure trees) (2009, methodology): “1(b) characterizes tightness for the stronger topology given in [27] based on certain L2-Wassersteinmetrics if one requires in addition uniform integrability of sampled mutual distance.”
Interpolated measures with bounded density in metric spaces satisfying the curvature-dimension conditions of Sturm (2011, methodology): “t one geodesic in the Wasserstein space along which all the functionals in a displacement convexity class DCN satisfy a convexity inequality. The second set of deﬁnitions, following the work of Sturm [20, 21], requires the same convexity inequality only for the critical entropy functionals. The thir”
On the heat flow on metric measure spaces: existence, uniqueness and stability (2010, methodology): “In [9] the definition was used in order to study the stability properties of Ricci curvature bounds, while in [12] the approach is slightly different.”
Comparison geometry for the Bakry-Emery Ricci tensor (2007, background): “Moreover Ricf has a natural extension to metric measure spaces [21, 35, 36].”
Geometry of manifolds with densities (2012, result): “ to the Euclidean space Rn with f(x) = 1 4 |x| 2 , one then recovers the classical logarithmic Sobolev inequality due to L. Gross [11]. More generally, in recentworkofLott and Villani [16], and Sturm [26, 27], a lower bound for the Bakry-Emery curvature is interpreted as a convexity measurement´ for”
A spectral notion of Gromov–Wasserstein distance and related methods (2011, methodology): “By exploiting the connections of Ricci curvature with mass transportation concepts, the work of Lott and Villani [49], Sturm [50] and Ollivier [51] has expanded considerably the understanding of how curvature may be defined on metric measure spaces and how it may play a role in guaranteeing compactn”