A volume comparison theorem and number of ends for manifolds with asymptotically nonnegative Ricci curvature (Q5933896)

The author considers a complete open Riemannian manifold \(M\) of asymptotically non-negative Ricci curvature. He shows that (1) the ratio of volumes vol\((B(x, R)\) and vol\((B(x, r)\) (where \(0 < r\leq R\) and \(x\in M\)) is bounded from above by a rational (given explicitly in the paper) function of \(R\) and \(r\), (2) \(M\) has finite number of ends and (3) the volume vol\(B(p R)\) (\(p\) being a base point in \(M\)) is bounded from below by \(C(\log R)^\rho\) for some positive constants \(C\) and \(\rho\).