Aspects and Examples on Quantitative Stratification with Lower Curvature Bounds

2020 ◽  
pp. 326-351
Author(s):  
Nan Li
Keyword(s):  
Curvature Bounds

scholarly journals Remarks on Manifolds with Two-Sided Curvature Bounds

2021 ◽  
Vol 9 (1) ◽  
pp. 53-64
Author(s):  
Vitali Kapovitch ◽  
Alexander Lytchak
Keyword(s):  
Distance Functions ◽  
Cut Locus ◽  
Bounded Curvature ◽  
Curvature Bounds ◽  
Positive Reach

Abstract We discuss folklore statements about distance functions in manifolds with two-sided bounded curvature. The topics include regularity, subsets of positive reach and the cut locus.

scholarly journals Smoothing Riemannian metrics with Ricci curvature bounds

1996 ◽  
Vol 90 (1) ◽  
pp. 49-61 ◽  
Author(s):  
Xianzhe Dai ◽  
Guofang Wei ◽  
Rugang Ye
Keyword(s):  
Ricci Curvature ◽  
Riemannian Metrics ◽  
Curvature Bounds

scholarly journals A∞-weights and compactness of conformal metrics under Ln∕2 curvature bounds

2021 ◽  
Vol 14 (7) ◽  
pp. 2163-2205
Author(s):  
Clara L. Aldana ◽  
Gilles Carron ◽  
Samuel Tapie
Keyword(s):  
Conformal Metrics ◽  
Curvature Bounds

scholarly journals Non-commutative Calculus, Optimal Transport and Functional Inequalities in Dissipative Quantum Systems

2019 ◽  
Vol 178 (2) ◽  
pp. 319-378 ◽  
Author(s):  
Eric A. Carlen ◽  
Jan Maas
Keyword(s):  
Spectral Gap ◽  
Optimal Transport ◽  
Quantum Systems ◽  
Sobolev Inequalities ◽  
Unified Framework ◽  
Logarithmic Sobolev Inequalities ◽  
Curvature Bounds ◽  
Finite Dimensional ◽  
Dissipative Quantum Systems ◽  
Entropy Inequalities

AbstractWe study dynamical optimal transport metrics between density matrices associated to symmetric Dirichlet forms on finite-dimensional $$C^*$$ C ∗ -algebras. Our setting covers arbitrary skew-derivations and it provides a unified framework that simultaneously generalizes recently constructed transport metrics for Markov chains, Lindblad equations, and the Fermi Ornstein–Uhlenbeck semigroup. We develop a non-nommutative differential calculus that allows us to obtain non-commutative Ricci curvature bounds, logarithmic Sobolev inequalities, transport-entropy inequalities, and spectral gap estimates.

Curvature Bounds for the Spectrum of Closed Einstein Spaces

1978 ◽  
Vol 30 (5) ◽  
pp. 1087-1091 ◽  
Author(s):  
Udo Simon
Keyword(s):  
Lower Bound ◽  
Scalar Curvature ◽  
Sectional Curvature ◽  
Constant Scalar Curvature ◽  
Einstein Space ◽  
Curvature Bounds ◽  
Nonzero Eigenvalue ◽  
Einstein Spaces ◽  
Image Position

The following is our main result.(A) THEOREM. Let (M, g) be a closed connected Einstein space, n = dim M ≧ 2 (with constant scalar curvature R). Let K0 be the lower bound of the sectional curvature. Then either (M, g) is isometrically diffeomorphic to a sphere and the first nonzero eigenvalue ƛ1of the Laplacian fulfils

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