Asymptotic volume and asymptotic isoperimetric profile of tori (Q2709138)

If \((T^n,g)\) is an \(n\)-dimensional Riemannian torus, its \textit{asymptotic volume} \(\text{VA}(T)\) is defined as the limit \(\lim_{r\to\infty} r^{-n}\text{vol}(B(x,r))\), where \(B(x,r)\) is a metric ball in the universal Riemannian covering \(\widetilde{T}\) of \(T\), and \(x\) is an arbitrary point in \(\widetilde{T}\). The asymptotic isoperimetric profile constant \(c_{\infty}(g)\) of the torus is the limit \(\lim_{\tau\to\infty} \tau^{-(n-1)/n} I(\tau)\), where \(I\) is the isoperimetric profile of \(\widetilde{T}\). In this exposition the author reviews some interesting results by \textit{D. Burago} and \textit{S. Ivanov} [Geom. Funct. Anal. 5, No. 5, 800-808 (1995; Zbl 0846.53043), ibid. 8, 783-787 (1998)], and \textit{P. Pansu} [ESAIM, Control Optim. Calc. Var. 4, 631-665 (1999; Zbl 0939.53022)] on these concepts.NEWLINENEWLINEFor the entire collection see [Zbl 0955.00015].