On asymptotic volume of tori
type: Item
instance of: scholarly article
title: On asymptotic volume of tori
zbMATH Open document ID: 0846.53043
DOI: 10.1007/BF01897051
published in: Geometric and Functional Analysis. GAFA
publication date: 1996-09-22T00:00:00Z
full work available at URL: https://eudml.org/doc/58210
review text: The authors investigate the asymptotic behaviour of volume of tori in Riemannian spaces. Given a \(\mathbb{Z}^n\)-periodic metric, the main term of the volume of a ball of radius \(R\), when \(R \to \infty\), in such a metric, is \(cR^n\) for some \(c > 0\). The authors prove that \(c \geq c_E\), where \(c_E\) is the constant for a flat metric, and equality holds if and only if the metric is flat. The proof is based on two lemmas. The first one generalizes a construction given by \textit{I. K. Babenko} [Math. Notes 44, No. 1/2, 579-586 (1988); translation from Mat. Zametki, 44, No. 2, 177-190 (1988; Zbl 0666.52005)], and, geometrically, it is equivalent to say that for any symmetric convex body \(K\) there exist supporting hyperplanes such that there exists an ellipsoid tangent to such hyperplanes which is inscribed in \(K\). By this lemma the authors obtain estimates of the norm with \(K\) as unit ball. The second lemma provides a generalized Besicovitch inequality. They also prove that if a sequence \(d_k\) of Riemannian metrics on a smooth manifold \(M\) converges to a Finsler metric \(d\) (the convergence is uniform on compact subsets of \(M\times M)\), then the volume is not increasing, i.e. \(\text{Vol} (M,d) \leq \lim \inf \text{Vol} (M,d_k)\) and equality implies that \(d\) is a Riemannian metric. The authors add some suggestions for further studies of this problem in Finsler spaces.
Mathematics Subject Classification ID: 53C45
Mathematics Subject Classification ID: 52A21
zbMATH DE Number: 847577
zbMATH Keywords: convergence of Riemannian metrics
zbMATH Keywords: asymptotic behaviour
zbMATH Keywords: volume of tori
zbMATH Keywords: supporting hyperplanes
zbMATH Keywords: Finsler spaces
author: Dmitri Burago
author: Sergei Ivanov
reviewed by: Salvatore Vassallo
MaRDI profile type: Publication
cites work: Q3799263
cites work: Riemannian tori without conjugate points are flat
cites work: VOLUMES OF BALLS IN MANIFOLDS WITHOUT CONJUGATE POINTS
cites work: A volume-diameter inequality for n-cubes
cites work: Filling Riemannian manifolds
OpenAlex ID: W1987062641
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description: scientific article; zbMATH DE number 847577