Volume growth and the topology of manifolds with nonnegative Ricci curvature
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.
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