scholarly journals Poincaré, modified logarithmic Sobolev and isoperimetric inequalities for Markov chains with non-negative Ricci curvature

2018 ◽  
Vol 274 (11) ◽  
pp. 3056-3089 ◽  
Author(s):  
Matthias Erbar ◽  
Max Fathi
Keyword(s):  
Markov Chains ◽  
Ricci Curvature ◽  
Isoperimetric Inequalities

scholarly journals Almost Euclidean Isoperimetric Inequalities in Spaces Satisfying Local Ricci Curvature Lower Bounds

2018 ◽  
Vol 2020 (5) ◽  
pp. 1481-1510 ◽  
Author(s):  
Fabio Cavalletti ◽  
Andrea Mondino
Keyword(s):  
Riemannian Manifold ◽  
Isoperimetric Inequality ◽  
Ricci Curvature ◽  
Lower Bounds ◽  
Ricci Flow ◽  
General Framework ◽  
Isoperimetric Inequalities ◽  
Optimal Transportation ◽  
Maximal Volume ◽  
Bounded Below

Abstract Motivated by Perelman’s Pseudo-Locality Theorem for the Ricci flow, we prove that if a Riemannian manifold has Ricci curvature bounded below in a metric ball which moreover has almost maximal volume, then in a smaller ball (in a quantified sense) it holds an almost euclidean isoperimetric inequality. The result is actually established in the more general framework of non-smooth spaces satisfying local Ricci curvature lower bounds in a synthetic sense via optimal transportation.

scholarly journals Eigenvalue estimates and isoperimetric inequalities for cone-manifolds

1993 ◽  
Vol 47 (1) ◽  
pp. 127-143 ◽  
Author(s):  
Craig Hodgson ◽  
Johan Tysk
Keyword(s):  
Ricci Curvature ◽  
Lower Bounds ◽  
Upper Bounds ◽  
Isoperimetric Inequalities ◽  
Eigenvalue Bounds ◽  
Eigenvalue Estimates ◽  
Maximal Diameter ◽  
Cone Type ◽  
Eigenvalues Of The Laplacian ◽  
Definition Of

This paper studies eigenvalue bounds and isoperimetric inequalities for Rieman-nian spaces with cone type singularities along a codimension-2 subcomplex. These “cone-manifolds” include orientable orbifolds, and singular geometric structures on 3-manifolds studied by W. Thurston and others.We first give a precise definition of “cone-manifold” and prove some basic results on the geometry of these spaces. We then generalise results of S.-Y. Cheng on upper bounds of eigenvalues of the Laplacian for disks in manifolds with Ricci curvature bounded from below to cone-manifolds, and characterise the case of equality in these estimates.We also establish a version of the Lévy-Gromov isoperimetric inequality for cone-manifolds. This is used to find lower bounds for eigenvalues of domains in cone-manifolds and to establish the Lichnerowicz inequality for cone-manifolds. These results enable us to characterise cone-manifolds with Ricci curvature bounded from below of maximal diameter.

Isoperimetric Inequalities and Decay of Iterated Kernels for Almost-transitive Markov Chains

1995 ◽  
Vol 4 (4) ◽  
pp. 419-442 ◽  
Author(s):  
L. Saloff-Coste
Keyword(s):  
Markov Chains ◽  
Random Walks ◽  
Isoperimetric Inequalities ◽  
Infinite Graphs ◽  
Decay Estimates ◽  
Finite Quotient

This paper gives precise isoperimetric inequalities for infinite graphs on which a group acts with finite quotient. Decay estimates are obtained for the iterated kernels of the associated random walks.

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