Gap theorems for hypersurfaces in \({\mathbb{R}}^ N\) (Q1070559)

In this paper the author studies noncompact properly imbedded hypersurfaces M of \({\mathbb{R}}^ N\), he gives sufficient conditions on M to be a hyperplane. These so-called Gap Theorems are similar to Bernstein's theorem in the theory of minimal surfaces. An essential quantity is the second fundamental form \(\alpha\) (p) of M and the value \(\tilde k(\)s)\(=\sup \{\| \alpha (p)\|^ 2\}\) on the ball \(| p| =s\). If \(\lim \inf_{s\to \infty}s^ 2\tilde k(s)=0\) and if further a convexity or curvature condition is fulfilled then M is a hyperplane.