scholarly journals Sobolev-to-Lipschitz property on $${\mathsf {QCD}}$$-spaces and applications

2021 ◽  
Author(s):  
Lorenzo Dello Schiavo ◽  
Kohei Suzuki
Keyword(s):  
Riemannian Manifolds ◽  
Heat Semigroup ◽  
Metric Measure Spaces ◽  
Lipschitz Property ◽  
Large Classes ◽  
Short Time Asymptotics ◽  
Time Asymptotics ◽  
Measure Spaces ◽  
Short Time ◽  
Appl Math

AbstractWe prove the Sobolev-to-Lipschitz property for metric measure spaces satisfying the quasi curvature-dimension condition recently introduced in Milman (Commun Pure Appl Math, to appear). We provide several applications to properties of the corresponding heat semigroup. In particular, under the additional assumption of infinitesimal Hilbertianity, we show the Varadhan short-time asymptotics for the heat semigroup with respect to the distance, and prove the irreducibility of the heat semigroup. These results apply in particular to large classes of (ideal) sub-Riemannian manifolds.

scholarly journals Bakry-Émery Conditions on Almost Smooth Metric Measure Spaces

2018 ◽  
Vol 6 (1) ◽  
pp. 129-145 ◽  
Author(s):  
Shouhei Honda
Keyword(s):  
Riemannian Manifolds ◽  
Short Note ◽  
Sufficient Condition ◽  
Metric Measure Spaces ◽  
Necessary And Sufficient Condition ◽  
Local Dimension ◽  
Base Points ◽  
Smooth Metric Measure Spaces ◽  
Measure Spaces ◽  
Necessary And Sufficient

Abstract In this short note, we give a sufficient condition for almost smooth compact metric measure spaces to satisfy the Bakry-Émery condition BE(K, N). The sufficient condition is satisfied for the glued space of any two (not necessary same dimensional) closed pointed Riemannian manifolds at their base points. This tells us that the BE condition is strictly weaker than the RCD condition even in this setting, and that the local dimension is not constant even if the space satisfies the BE condition with the coincidence between the induced distance by the Cheeger energy and the original distance. In particular, the glued space gives a first example with a Ricci bound from below in the Bakry-Émery sense, whose local dimension is not constant. We also give a necessary and sufficient condition for such spaces to be RCD(K, N) spaces.

scholarly journals Angles between Curves in Metric Measure Spaces

2017 ◽  
Vol 5 (1) ◽  
pp. 47-68
Author(s):  
Bang-Xian Han ◽  
Andrea Mondino
Keyword(s):  
Riemannian Manifolds ◽  
Metric Space ◽  
Optimal Transportation ◽  
Metric Measure Spaces ◽  
Alexandrov Spaces ◽  
Measure Spaces

Abstract The goal of the paper is to study the angle between two curves in the framework of metric (and metric measure) spaces. More precisely, we give a new notion of angle between two curves in a metric space. Such a notion has a natural interplay with optimal transportation and is particularly well suited for metric measure spaces satisfying the curvature-dimension condition. Indeed one of the main results is the validity of the cosine formula on RCD*(K, N) metric measure spaces. As a consequence, the new introduced notions are compatible with the corresponding classical ones for Riemannian manifolds, Ricci limit spaces and Alexandrov spaces.

scholarly journals Graph approximations to the Laplacian spectra

2020 ◽  
pp. 1-35
Author(s):  
Jinpeng Lu
Keyword(s):  
Riemannian Manifolds ◽  
Compact Riemannian Manifold ◽  
Neumann Boundary ◽  
Beltrami Operator ◽  
Metric Measure Spaces ◽  
Proximity Graphs ◽  
Manifold With Boundary ◽  
Graph Laplacians ◽  
Laplacian Spectra ◽  
Measure Spaces

I prove that the spectrum of the Laplace–Beltrami operator with the Neumann boundary condition on a compact Riemannian manifold with boundary admits a fast approximation by the spectra of suitable graph Laplacians on proximity graphs on the manifold, and similar graph approximation works for metric-measure spaces glued out of compact Riemannian manifolds of the same dimension.

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