On complete hypersurfaces with negative Ricci curvature in Euclidean spaces

Abstract

In this paper, we prove that if $M^n$, $n\ge 3$, is a complete Riemannian manifold with negative Ricci curvature and $f:M^n\to {\mathbb{R}}^{n+1}$ 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.