# The equality case in Cheeger's and Buser's inequalities on $\mathsf{RCD}$ spaces

created by mondino on 31 Aug 2020
modified by semola on 14 Apr 2021

[BibTeX]

Published Paper

Inserted: 31 aug 2020
Last Updated: 14 apr 2021

Journal: Journal of Functional Analysis
Volume: 281
Number: 3
Year: 2021
Doi: https://doi.org/10.1016/j.jfa.2021.109022

ArXiv: 2008.12358 PDF

Abstract:

We prove that the sharp Buser's inequality obtained in the framework of $\mathsf{RCD}(1,\infty)$ spaces by the first two authors is rigid, i.e. equality is obtained if and only if the space splits isomorphically a Gaussian. The result is new even in the smooth setting. We also show that the equality in Cheeger's inequality is never attained in the setting of $\mathsf{RCD}(K,\infty)$ spaces with finite diameter or positive curvature, and we provide several examples of spaces with Ricci curvature bounded below where these assumptions are not satisfied and the equality is attained.