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 |
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Springer Verlag
2020
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| _version_ | 1826262537329442816 |
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| author | De Ponti, N Mondino, A |
| author_facet | De Ponti, N Mondino, A |
| author_sort | De Ponti, N |
| collection | OXFORD |
| description | 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. |
| first_indexed | 2024-03-06T19:37:46Z |
| format | Journal article |
| id | oxford-uuid:1f9e6684-a046-4616-97ed-641e6d25c236 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-06T19:37:46Z |
| publishDate | 2020 |
| publisher | Springer Verlag |
| record_format | dspace |
| spelling | oxford-uuid:1f9e6684-a046-4616-97ed-641e6d25c2362022-03-26T11:22:54ZSharp Cheeger–Buser type inequalities in RCD(K,∞) spacesJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:1f9e6684-a046-4616-97ed-641e6d25c236EnglishSymplectic Elements at OxfordSpringer Verlag2020De Ponti, NMondino, AThe 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. |
| spellingShingle | De Ponti, N Mondino, A Sharp Cheeger–Buser type inequalities in RCD(K,∞) spaces |
| title | Sharp Cheeger–Buser type inequalities in RCD(K,∞) spaces |
| title_full | Sharp Cheeger–Buser type inequalities in RCD(K,∞) spaces |
| title_fullStr | Sharp Cheeger–Buser type inequalities in RCD(K,∞) spaces |
| title_full_unstemmed | Sharp Cheeger–Buser type inequalities in RCD(K,∞) spaces |
| title_short | Sharp Cheeger–Buser type inequalities in RCD(K,∞) spaces |
| title_sort | sharp cheeger buser type inequalities in rcd k ∞ spaces |
| work_keys_str_mv | AT depontin sharpcheegerbusertypeinequalitiesinrcdkspaces AT mondinoa sharpcheegerbusertypeinequalitiesinrcdkspaces |