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 $\frac{(-K)}{H^3}\ge c>0$ 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 $\frac{(-K)}{H^3}\ge c>0$ everywhere}. At last, it will be proved that, if there exists a stable hypersurface with zero scalar curvature and $\frac{(-K)}{H^3}\ge c>0$ everywhere, that is, with volume growth greater than polynomial, then its tubular neighborhood is not embedded for suitable radius.

25 pages, 1 figure. Minor corrections in Proposition 2.1. Published in Math. Zeitschrift June 2014, Volume 277, Issue 1-2, pp 481-497

Cited by in corpus (1)

• Stable hypersurfaces with zero scalar curvature in Euclidean space