Volume growth and the topology of manifolds with nonnegative Ricci curvature
type: Item
instance of: scholarly article
title: Volume growth and the topology of manifolds with nonnegative Ricci curvature
zbMATH Open document ID: 1200.53043
author: Michael Munn
published in: The Journal of Geometric Analysis
publication date: 2010-06-29T00:00:00Z
full work available at URL: https://arxiv.org/abs/0712.0827
review text: The paper under review gives explicit constants \(0<\alpha(k,n)<1\), so that if the volume growth of an \(n\)-dimensional manifold with non-negative Ricci curvature is larger than \(\alpha(k,n)\) then the \(k\)-th homotopy group of the manifold is trivial. Since this constant is increasing with \(k\), when the volume growth is larger than \(\alpha(n,n)\) then the manifold is contractible. For a manifold with non-negative Ricci curvature, the quotient between the volume of a ball of radius \(r\) and a Euclidean ball of the same radius, is non-increasing on the radius \(r\). When \(r\) goes to zero this quotient is one, and when \(r\) goes to infinity this is called the volume growth. The existence of those constants is due to \textit{G. Perelman} [J. Am. Math. Soc. 7, No.~2, 299--305 (1994; Zbl 0799.53050)], and here it is made explicit. The paper uses the techniques of Perelman.
reviewed by: Joan Porti
Mathematics Subject Classification ID: 53C21
Mathematics Subject Classification ID: 53C20
zbMATH DE Number: 5728280
zbMATH Keywords: nonnegative Ricci curvature
zbMATH Keywords: volume growth
zbMATH Keywords: homotopy groups
MaRDI profile type: Publication
OpenAlex ID: W2147011092
arXiv ID: 0712.0827
Wikidata QID: Q125585597
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DOI: 10.1007/S12220-010-9125-4
description: scientific article