Stability Properties for the Higher Dimensional Catenoid in R Luen-fai Tam and Detang Zhou

This paper concerns some stability properties of higher dimensional catenoids in Rn+1 with n ≥ 3. We prove that higher dimensional catenoids have index one. We use δ-stablity for minimal hypersurfaces and show that the catenoid is 2 n -stable and that a complete 2 n -stable minimal hypersurface is a catenoid or a hyperplane provided the second fundamental form satisfies some decay conditions.