Volume growth of some uniformly contractible Riemannian manifolds
A metric space \((M, d)\) is called uniformly contractible if, for every \(r> 0\), there exists \(R(r)\geq r\) such that every ball with radius \(r\) is contractible in the concentric ball with radius \(R(r)\). This paper deals with the volume growth of uniformly contractible Riemannian manifolds. More precisely, let \(BM_n\) be the Banach-Mazur space of \(n\)-dimensional Banach spaces, equipped with the Banach-Mazur distance. Given a metric space \(X\) denote by \(\text{ALDist}(X, BM_n)\) the asymptotic Lipschitz distance of \(X\) and \(BM_n\). The author proves that, if \((M^n, g)\) is a uniformly contractible Riemannian manifold, such that \(\text{ALDist}(M, BM_n)\leq k\) \((k\geq 1)\), then \[ \liminf_{R\to\infty}\,{\text{Vol}_g(B(R))\over R^n\omega_n}\geq {1\over K^{2n}}, \] where \(B(R)\) is the ball of radius \(R\) centered at any point of \(M\) and \(\omega_n\) is the volume of the unit Euclidean ball. It follows that an uniformly contractible Riemannian manifold \((M^n, g)\) has at least Euclidean volume growth, provided that there exists a Banach space \((V^n,\|\;\|)\) such that the Gromov-Hausdorff distance satisfies: \(d_{GH}(M, V^n)<\infty\). This gives a solution to a problem posed by \textit{M. Gromov} in [(*) Metric structures for Riemannian and non-Riemannian spaces. Progress in Mathematics. 152, Birkäuser, Boston (1999; Zbl 0953.53002)] and covers an earlier result due to Burago and Ivanov. Finally, the author proves a generalization of a volume growth theorem stated in [(*)].
scientific article; zbMATH DE number 5133232
scientific article