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.

19 pages. Final version published in The Journal of Geometric Analysis