A Remark on the Omori-Yau Maximum Principle

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A Riemannian manifold M is said to satisfy the Omori-Yau maximum principle if for any C² bounded function g:M→ R there is a sequence xₙ∈ M, such that ₙ→ ∞g(xₙ)=ₘ g, ₙ→ ∞|∇ g(xₙ)|=0 and ₙ→ ∞Δg(xₙ)≤ 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.