On Stable Hypersurfaces with Vanishing Scalar Curvature

Abstract

We will prove that \emph{there are no stable complete hypersurfaces of $\mathbb{R}^4$ with zero scalar curvature, polynomial volume growth and such that $\dfrac{(-K)}{H^3}\geq c>0$ everywhere, for some constant $c>0$}, where $K$ denotes the Gauss-Kronecker curvature and $H$ denotes the mean curvature of the immersion. Our second result is the Bernstein type one \emph{there is no entire graphs of $\mathbb{R}^4$ with zero scalar curvature such that $\dfrac{(-K)}{H^3}\geq c>0$ everywhere}. At last, it will be proved that, if there exists a stable hypersurface with zero scalar curvature and $\dfrac{(-K)}{H^3}\geq c>0$ everywhere, that is, with volume growth greater than polynomial, then its tubular neighborhood is not embedded for suitable radius.