Sharp Cheeger-Buser type inequalities in $\mathsf{RCD}(K,\infty )$ spaces

Abstract

The goal of the paper is to sharpen and generalise bounds involving the Cheeger's isoperimetric constant $h$ and the first eigenvalue ${\lambda }_1$ of the Laplacian. A celebrated lower bound of ${\lambda }_1$ in terms of $h$, ${\lambda }_1\ge h^2/4$, was proved by Cheeger in 1970 for smooth Riemannian manifolds. An upper bound on ${\lambda }_1$ 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 two fold. 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 $K\in \mathbb{R}$ (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 $\mathsf{RCD}(K,\infty )$ spaces.

arXiv comment19 pages. Final version published in The Journal of Geometric Analysis

References in corpus (9)

• Metric measure spaces with Riemannian Ricci curvature bounded from below
• Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below
• Riemannian Ricci curvature lower bounds in metric measure spaces with $\sigma$-finite measure
• Self-improvement of the Bakry-Émery condition and Wasserstein contraction of the heat flow in RCD(K,\infty) metric measure spaces
• Sharp and rigid isoperimetric inequalities in metric-measure spaces with lower Ricci curvature bounds
• Sharp geometric and functional inequalities in metric measure spaces with lower Ricci curvature bounds
• Polya-Szego inequality and Dirichlet $p$-spectral gap for non-smooth spaces with Ricci curvature bounded below
• Gaussian-type Isoperimetric Inequalities in $RCD(K,\infty )$ probability spaces for positive $K$
• Isoperimetric inequalities for finite perimeter sets under lower Ricci curvature bounds

Cited by in corpus (5)

• Gravity from thermodynamics: optimal transport and negative effective dimensions
• The Cheeger problem in abstract measure spaces
• The equality case in Cheeger's and Buser's inequalities on $\mathsf{RCD}$ spaces
• Higher Cheeger ratios of features in Laplace-Beltrami eigenfunctions
• Cheeger bounds on spin-two fields