The equality case in Cheeger's and Buser's inequalities on $\mathsf{RCD}$ spaces
Functional Analysis
2022-11-11 v2 Differential Geometry
Metric Geometry
Abstract
We prove that the sharp Buser's inequality obtained in the framework of 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 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.
Cite
@article{arxiv.2008.12358,
title = {The equality case in Cheeger's and Buser's inequalities on $\mathsf{RCD}$ spaces},
author = {Nicolò De Ponti and Andrea Mondino and Daniele Semola},
journal= {arXiv preprint arXiv:2008.12358},
year = {2022}
}
Comments
Added new results: the discussion on Cheeger's inequality now fits into the study of a family of inequalities relating eigenvalues of the p-Laplacian. To appear on Journal of Functional Analysis