Ball covering on manifolds with nonnegative Ricci curvature near infinity

Abstract:

Let $M$ be a complete open Riemannian manifold with nonnegative Ricci curvature outside a compact set $B$. We show that the following ball covering property (see [LT]) is true provided that the sectional curvature has a lower bound: For a fixed ${p_0} \in M$, there exist $N > 0$ and ${r_0} > 0$ such that for $r \geq {r_0}$, there exist $p_1$, …, $p_k \in \partial {B_{p0}}(2r),k \leq N$ , with \[ \bigcup \limits _{j = 1}^k {{B_{{p_j}}}(r) \supset \partial {B_{{p_0}}}(2r).} \] Furthermore $N$ and ${r_0}$ depend only on the dimension, the lower bound on the sectional curvature, and the radius of the ball at ${p_0}$ that contains $B$.