On the asymptotic isoperimetric constant of tori
Let \((M,g)\) be the universal cover of a Riemannian \(n\)-torus. The asymptotic isoperimetric constant of \((M,g)\) is \[ \sigma(M,g)=\lim\sup_{\text{Vol}_n(\Omega)\to \infty} \text{Vol}_n(\Omega,g)^{1/2}/ \text{Vol}_{n-1}(\partial\Omega,g)^{1/{n-1}}, \] where \(\text{Vol}_n\) and \(\text{Vol}_{n-1}\) are Riemannian measures (for \(g\)) of the respective dimensions and \(\Omega\) ranges over all open bounded subsets of \(M\). Theorem: If \(n\geq 3\), then there exist \({\mathbb Z}^n\)-periodic Riemannian metrics on \({\mathbb R}^n\) with arbitrarily small asymptotic constants. This is a continuation of the authors' earlier work [ibid. 5, 800-808 (1995; Zbl 0846.53043)].
scientific article; zbMATH DE number 1223297