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