Volume growth and the topology of manifolds with nonnegative Ricci curvature

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

2007-12-05

arxiv.org/abs/0712.0827v3

Mathematics

Differential Geometry

v3 is based on the earlier v1. The current version simplifies the argument of Lemma 2.4 and corrects some minor typos. We also

Scientific paper

Let $M^n$ be a complete, open Riemannian manifold with $\Ric \geq 0$. In 1994, Grigori Perelman showed that there exists a constant $\delta_{n}>0$, depending only on the dimension of the manifold, such that if the volume growth satisfies $\alpha_M := \lim_{r \to \infty} \frac{\Vol(B_p(r))}{\omega_n r^n} \geq 1-\delta_{n}$, then $M^n$ is contractible. Here we employ the techniques of Perelman to find specific lower bounds for the volume growth, $\alpha(k,n)$, depending only on $k$ and $n$, which guarantee the individual $k$-homotopy group of $M^n$ is trivial.

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