A remark on the Omori-Yau maximum principle

A Riemannian manifold M is said to satisfy the Omori-Yau maximum principle if for any C^2 bounded function g: M rightarrow R there is a sequence x n ε M, such that lim n rightarrow ∞ g(xn) = sup Mg, lim n rightarrow ∞ |triangledown g(x n)| = 0 and limsup n rightarrow ∞ ∆g(x n) ≤ 0. It is shown that if the Ricci curvature does not approach -∞ too fast the manifold satisfies the Omori-Yau maximum principle. This improves earlier necessary conditions. The given condition is quite optimal