Sharp Cheeger–Buser type inequalities in RCD(K,∞) spaces
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...
| Main Authors: | , |
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| Format: | Journal article |
| Language: | English |
| Published: |
Springer Verlag
2020
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| Summary: | 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. |
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