论文标题
Cheeger's和Buser的不平等现象中的平等情况
The equality case in Cheeger's and Buser's inequalities on $\mathsf{RCD}$ spaces
论文作者
论文摘要
我们证明，在$ \ mathsf {rcd}（1，1，\ infty）$空间的框架中获得的尖锐的Buser的不平等是刚性的，即仅在空间拆分同样的高斯时，才能获得平等。即使在平滑的设置中，结果也是新的。我们还表明，Cheeger不平等的平等在$ \ Mathsf {rcd}（k，\ infty）$空间的情况下，直径有限或正曲率的空间，我们提供了几个带有RICCI曲率的示例，而RICCI曲率下面限制了下面这些假设并不满足等于等于的假设。
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.
