Resumen de On complete hypersurfaces with negative Ricci curvature in Euclidean spaces

Alexandre Paiva, Francisco Fontenele

• In this paper, we prove that if Mn, n≥3, is a complete Riemannian manifold with negative Ricci curvature and f:Mn→Rn+1 is an isometric immersion such that Rn+1\f(M) is an open set that contains balls of arbitrarily large radius, then infM∣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 Rn+1. This gives a partial answer to a question raised by Reilly and Yau.