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We prove that the hypotheses in the Pigola-Rigoli-Setti version of the Omori-Yau maximum principle are logically equivalent to the assumption that the manifold carries a C 2 proper function whose gradient and Hessian (Laplacian) are bounded. In particular, this result extends the scope of the original Omori-Yau principle, formulated in terms of lower bounds for curvature.2010 Mathematics subject classification: primary 53C21; secondary 35B50.

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In this paper, we prove that if M^n , n\geq 3 , is a complete Riemannian manifold with negative Ricci curvature and f\colon M^n\to\mathbb{R}^{n+1} is an isometric immersion such that \mathbb{R}^{n+1}\backslash f(M) is an open set that contains balls of arbitrarily large radius, then \inf_M|A|=0 , where |A| is the norm of the second fundamental form of the immersion. In particular, an n -dimensional complete Riemannian manifold with negative Ricci curvature bounded away from zero cannot be properly isometrically immersed in a half-space of \mathbb{R}^{n+1} . This gives a partial answer to a question raised by Reilly and Yau.

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