Asymptotic volume and asymptotic isoperimetric profile of tori. (Volume et profil isopérimétrique asymptotiques des tores.) (French) Zbl 0994.53016

Séminaire de théorie spectrale et géométrie. Année 1999-2000. St. Martin d’Hères: Université de Grenoble I, Institut Fourier, Sémin. Théor. Spectr. Géom. 18, 43-47 (2000).
If \((T^n,g)\) is an \(n\)-dimensional Riemannian torus, its 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 D. Burago and S. Ivanov [Geom. Funct. Anal. 5, No. 5, 800-808 (1995; Zbl 0846.53043), ibid. 8, 783-787 (1998)], and P. Pansu [ESAIM, Control Optim. Calc. Var. 4, 631-665 (1999; Zbl 0939.53022)] on these concepts.
For the entire collection see [Zbl 0955.00015].


53C20 Global Riemannian geometry, including pinching
53C60 Global differential geometry of Finsler spaces and generalizations (areal metrics)
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