Volume growth of some uniformly contractible Riemannian manifolds
type: Item
instance of: scholarly article
title: Volume growth of some uniformly contractible Riemannian manifolds
zbMATH Open document ID: 1122.53021
author: Luofei Liu
published in: Mathematische Zeitschrift
publication date: 2007-03-13T00:00:00Z
review text: 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 [(*)].
reviewed by: Maria Falcitelli
Mathematics Subject Classification ID: 53C23
Mathematics Subject Classification ID: 53C20
zbMATH DE Number: 5133232
zbMATH Keywords: volume growth
zbMATH Keywords: uniformly contractible space
MaRDI profile type: Publication
full work available at URL: s00209-006-0038-7
OpenAlex ID: W2079381794
cites work: Smooth approximation of Lipschitz functions on Riemannian manifolds
cites work: Boundary case of equality in optimal Loewner-type inequalities
cites work: Q4026493
cites work: On asymptotic volume of tori
cites work: On asymptotic volume of Finsler tori, minimal surfaces in normed spaces, and symplectic filling volume.
cites work: A course in metric geometry
cites work: Q4039749
cites work: On the Higson-Roe corona
cites work: Large Riemannian manifolds which are flexible
cites work: Uniform structures of metric spaces and open manifolds
cites work: Quasi-flats and rigidity in higher rank symmetric spaces
cites work: Approximating Topological Metrics by Riemannian Metrics
cites work: \(C^\infty\) convex functions and manifolds of positive curvature
cites work: $C^\infty$ approximations of convex, subharmonic, and plurisubharmonic functions
cites work: Filling Riemannian manifolds
cites work: Q3736498
cites work: Q3140233
cites work: Generalized degree and optimal Loewner-type inequalities
cites work: Asymptotic theory of finite dimensional normed spaces. With an appendix by M. Gromov: Isoperimetric inequalities in Riemannian manifolds
cites work: Q3994358
Wikidata QID: Q115388825
DOI: 10.1007/S00209-006-0038-7
Recommended article: Closed geodesics and volume growth of Riemannian manifolds
Recommended article: A note on the nullity of homotopy group for complete three dimensional manifolds with Ricci \(\geq 0\)
Recommended article: The almost rigidity of manifolds with lower bounds on Ricci curvature and minimal volume growth
Recommended article: Q4344577
Recommended article: Ricci curvature and the topology of open manifolds
Recommended article: Sharp geometric inequalities for closed hypersurfaces in manifolds with nonnegative Ricci curvature
Recommended article: Large manifolds with positive Ricci curvature
Recommended article: Sur la croissance du volume dans une classe conforme. (On the volume growth in a conformal class)
Recommended article: Little topology, big volume
Recommended article: The ends of manifolds with bounded geometry, linear growth and finite filling area
description: scientific article; zbMATH DE number 5133232