scholarly journals Complete logarithmic Sobolev inequalities via Ricci curvature bounded below

2022 ◽  
Vol 394 ◽  
pp. 108129
Author(s):  
Michael Brannan ◽  
Li Gao ◽  
Marius Junge
Keyword(s):  
Ricci Curvature ◽  
Sobolev Inequalities ◽  
Logarithmic Sobolev Inequalities ◽  
Bounded Below

ON THE COMPACTNESS OF MANIFOLDS

2003 ◽  
Vol 06 (supp01) ◽  
pp. 29-38 ◽  
Author(s):  
XUE-MEI LI ◽  
FENG-YU WANG
Keyword(s):  
Ricci Curvature ◽  
Riemannian Manifolds ◽  
Positive Constant ◽  
Riemannian Metric ◽  
Sobolev Inequalities ◽  
Functional Inequalities ◽  
The Family ◽  
Bounded Below

It is believed that the family of Riemannian manifolds with negative curvatures is much richer than that with positive curvatures. In fact there are many results on the obstruction of furnishing a manifold with a Riemannian metric whose curvature is positive. In particular any manifold admitting a Riemannian metric whose Ricci curvature is bounded below by a positive constant must be compact. Here we investigate such obstructions in terms of certain functional inequalities which can be considered as generalized Poincaré or log-Sobolev inequalities. A result of Saloff-Coste is extended.

BEYOND STRONG SUBADDITIVITY? IMPROVED BOUNDS ON THE CONTRACTION OF GENERALIZED RELATIVE ENTROPY

1994 ◽  
Vol 06 (05a) ◽  
pp. 1147-1161 ◽  
Author(s):  
MARY BETH RUSKAI
Keyword(s):  
Relative Entropy ◽  
Discrete Case ◽  
Sobolev Inequalities ◽  
Completely Positive ◽  
Logarithmic Sobolev Inequalities ◽  
Strong Subadditivity

New bounds are given on the contraction of certain generalized forms of the relative entropy of two positive semi-definite operators under completely positive mappings. In addition, several conjectures are presented, one of which would give a strengthening of strong subadditivity. As an application of these bounds in the classical discrete case, a new proof of 2-point logarithmic Sobolev inequalities is presented in an Appendix.

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