Sharp Cheeger–Buser type inequalities in RCD(K,∞) spaces

Journal article

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.

Authors: De Ponti, N; Mondino, A

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Springer Verlag
• Journal: Journal of Geometric Analysis
• Volume: 31
• Pages: 2416–2438
• Publication date: 2020-02-14
• Acceptance date: 2020-01-12
• DOI: 10.1007/s12220-020-00358-6
• EISSN: 1559-002X
• ISSN: 1050-6926
• Language: English
• Keywords: Ricci curvature; Buser inequality; Cheeger inequality; first eigenvalue laplace operator; metric measure spaces; FFR