Volume growth and the topology of manifolds with nonnegative Ricci curvature

TL;DR

This paper investigates how the volume growth of nonnegatively Ricci curved manifolds influences their topology, providing explicit bounds that ensure triviality of specific homotopy groups.

Contribution

It extends Perelman's results by establishing explicit volume growth thresholds that guarantee the triviality of individual homotopy groups in such manifolds.

Findings

Derived lower bounds for volume growth depending on dimension and homotopy group index.

Proved that exceeding these bounds implies certain topological trivialities.

Connected volume growth rates with topological properties of manifolds.

Abstract

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.