On Stable Hypersurfaces with Vanishing Scalar Curvature

TL;DR

This paper proves non-existence results for stable hypersurfaces and entire graphs in ^4 with zero scalar curvature under certain curvature and volume growth conditions, extending understanding of geometric stability.

Contribution

It establishes new non-existence theorems for stable hypersurfaces and entire graphs with zero scalar curvature and specific curvature bounds in ^4.

Findings

No stable complete hypersurfaces with zero scalar curvature and polynomial volume growth exist under the given curvature ratio condition.

There are no entire graphs in ^4 with zero scalar curvature satisfying the curvature ratio condition.

If such a hypersurface with volume growth exceeding polynomial exists, its tubular neighborhood cannot be embedded.

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.