On asymptotic volume of tori. (English) Zbl 0846.53043

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 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.


53C45 Global surface theory (convex surfaces à la A. D. Aleksandrov)
52A21 Convexity and finite-dimensional Banach spaces (including special norms, zonoids, etc.) (aspects of convex geometry)
Full Text: DOI EuDML


[1] [B]I. Babenko, Asymptotic volume of tori and geometry of convex bodies, Mat. Zametki 44:2 (1988), 177–188. · Zbl 0653.52008
[2] [Bu1]D. Burago, Periodic metrics, Advances in Soviet Math., New York 9 (1992), 205–210.
[3] [Bu2]D. Burago, Periodic metrics, in ”Seminar on Dynamical Systems”, Progress in Nonlinear Differential Equations (H. Brezis, ed.), Birkhäuser Verlag, 12 (1994), 90–96.
[4] [BuI]D. Burago, S. Ivanov, Riemannian tori without conjugate points are flat, GAFA 4:3 (1994), 259–269. · Zbl 0808.53038 · doi:10.1007/BF01896241
[5] [BuZ]Yu. Burago, V. Zalgaller, Geometric Inequalities, Springer-Verlag (1988).
[6] [C]C. Croke, Volumes of balls in manifolds without conjugate points, Int. J. Math. 3:4 (1992), 455–467. · Zbl 0772.53031 · doi:10.1142/S0129167X92000205
[7] [D1]W. Derrick, A weighted volume-diameter inequality forn-cube, J. Math. Mech. 18:5 (1968), 455–467.
[8] [D2]W. Derrick, A volume-diameter inequality forn-cube, Analysis Math. 22 (1969), 1–36. · Zbl 0177.30702 · doi:10.1007/BF02786781
[9] [G1]M. Gromov, Filling Riemannian manifolds, J. Diff. Geom. 18 (1983), 1–147. · Zbl 0515.53037
[10] [G2]M. Gromov, Dimension, non-linear spectra and width, Springer Lecture Notes in Mathematics 1317 (1988), 132–184. · doi:10.1007/BFb0081739
[11] [G3]M. Gromov, Asymptotic Invariants of Infinite Groups, ”Geometric Group Theory vol 2”, London Math. Society Lecture Notes 182 (1993).
[12] [GLP]M. Gromov, J. Lafontaine, P. Pansu, Structure metriques pour les varietés Riemanniennes, CEDIC/Fernand Math, Paris 1981.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.