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

TL;DR

This paper characterizes the conditions under which equality holds in Cheeger's and Buser's inequalities within $ ext{RCD}$ spaces, revealing rigidity results and providing new insights even in smooth cases.

Contribution

It establishes the rigidity of Buser's inequality in $ ext{RCD}(1, olinebreak\infty)$ spaces and shows that equality in Cheeger's inequality is not attained under certain curvature conditions.

Findings

Equality in Buser's inequality implies the space splits as a Gaussian.

Equality in Cheeger's inequality is never attained in certain $ ext{RCD}$ spaces.

Examples of spaces where these inequalities are sharp or not attained are provided.

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.