$\mathsf{RCD}$空间上Cheeger不等式和Buser不等式的相等情形
The equality case in Cheeger's and Buser's inequalities on $\mathsf{RCD}$ spaces
摘要：
本文证明了前两个作者在$\mathsf{RCD}(1,\infty)$空间框架下得到的Sharp Buser不等式是刚性的，即等式成立的充要条件是空间同构地分裂为Gauss空间.结果是新的，即使在顺利设置。我们还表明，在Cheeger不等式的平等从来没有达到的设置$\mathsf{RCD}(K,\infty)$空间有限直径或正曲率，我们提供了几个例子的空间与里奇曲率有界以下这些假设不满足和平等的实现。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
