Calculus of Variations and Geometric Measure Theory

N. De Ponti - A. Mondino

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

created by mondino on 11 Feb 2019
modified on 10 Feb 2020


Published Paper

Inserted: 11 feb 2019
Last Updated: 10 feb 2020

Journal: The Journal of Geometric Analysis
Year: 2020
Doi: 10.1007/s12220-020-00358-6


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 $h$, $\lambda_{1}\geq h^{2}/4$, 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-\'Emery 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 $RCD(K,\infty)$ spaces.