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

This paper concerns some stability properties of higher dimensional catenoids in ${\text{rr}}^{n+1}$ 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 $\frac{2}{n}$-stable and a complete $\frac{2}{n}$-stable minimal hypersurface is a catenoid or a hyperplane provided the second fundamental form satisfies some decay conditions.