We prove that the sharp Buser's inequality obtained in the framework of RCD(1,∞) spaces by the first two authors [29] 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 RCD(K,∞) 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.
De Ponti, N., Mondino, A., Semola, D. (2021). The equality case in Cheeger's and Buser's inequalities on RCD spaces. JOURNAL OF FUNCTIONAL ANALYSIS, 281(3) [10.1016/j.jfa.2021.109022].
The equality case in Cheeger's and Buser's inequalities on RCD spaces
De Ponti, N
;
2021
Abstract
We prove that the sharp Buser's inequality obtained in the framework of RCD(1,∞) spaces by the first two authors [29] 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 RCD(K,∞) 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.
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