The goal of the paper is to sharpen and generalise bounds involving Cheeger’s isoperimetric constant h and the first eigenvalue λ1 of the Laplacian. A celebrated lower bound of λ1 in terms of h, λ1≥ h2/ 4 , was proved by Cheeger in 1970 for smooth Riemannian manifolds. An upper bound on λ1 in terms of h was established by Buser in 1982 (with dimensional constants) and improved (to a dimension-free estimate) by Ledoux in 2004 for smooth Riemannian manifolds with Ricci curvature bounded below. The goal of the paper is twofold. First: we sharpen the inequalities obtained by Buser and Ledoux obtaining a dimension-free sharp Buser inequality for spaces with (Bakry–Émery weighted) Ricci curvature bounded below by K∈ R (the inequality is sharp for K> 0 as equality is obtained on the Gaussian space). Second: all of our results hold in the higher generality of (possibly non-smooth) metric measure spaces with Ricci curvature bounded below in synthetic sense, the so-called RCD(K, ∞) spaces.

De Ponti, N., Mondino, A. (2021). Sharp Cheeger–Buser Type Inequalities in RCD(K, ∞) Spaces. THE JOURNAL OF GEOMETRIC ANALYSIS, 31(3), 2416-2438 [10.1007/s12220-020-00358-6].

Sharp Cheeger–Buser Type Inequalities in RCD(K, ∞) Spaces

De Ponti, N;
2021

Abstract

The goal of the paper is to sharpen and generalise bounds involving Cheeger’s isoperimetric constant h and the first eigenvalue λ1 of the Laplacian. A celebrated lower bound of λ1 in terms of h, λ1≥ h2/ 4 , was proved by Cheeger in 1970 for smooth Riemannian manifolds. An upper bound on λ1 in terms of h was established by Buser in 1982 (with dimensional constants) and improved (to a dimension-free estimate) by Ledoux in 2004 for smooth Riemannian manifolds with Ricci curvature bounded below. The goal of the paper is twofold. First: we sharpen the inequalities obtained by Buser and Ledoux obtaining a dimension-free sharp Buser inequality for spaces with (Bakry–Émery weighted) Ricci curvature bounded below by K∈ R (the inequality is sharp for K> 0 as equality is obtained on the Gaussian space). Second: all of our results hold in the higher generality of (possibly non-smooth) metric measure spaces with Ricci curvature bounded below in synthetic sense, the so-called RCD(K, ∞) spaces.
Articolo in rivista - Articolo scientifico
Buser inequality; Cheeger inequality; First eigenvalue laplace operator; Metric measure spaces; Ricci curvature;
English
14-feb-2020
2021
31
3
2416
2438
none
De Ponti, N., Mondino, A. (2021). Sharp Cheeger–Buser Type Inequalities in RCD(K, ∞) Spaces. THE JOURNAL OF GEOMETRIC ANALYSIS, 31(3), 2416-2438 [10.1007/s12220-020-00358-6].
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10281/462759
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