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

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.

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

References in corpus (2)

• Rigidity for the spectral gap on $RCD(K,\infty )$-spaces
• From Harnack inequality to heat kernel estimates on metric measure spaces and applications

Cited by in corpus (3)

• Gravity from thermodynamics: optimal transport and negative effective dimensions
• The Cheeger problem in abstract measure spaces
• Cheeger bounds on spin-two fields