Stability properties for the higher dimensional catenoid in $\mathbb{R}^{n+1}$

This paper concerns some stability properties of higher dimensional catenoids in $\mathbb{R}^{n+1}$ with $n\ge 3$. We prove that higher dimensional catenoids have index one. We use $\delta$-stablity for minimal hypersurfaces and show that the catenoid is $\frac 2n$-stable and that a complete $\frac 2n$-stable minimal hypersurface is a catenoid or a hyperplane provided the second fundamental form satisfies some decay conditions.