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

TL;DR

This paper investigates the stability characteristics of higher-dimensional catenoids in Euclidean space, establishing their index, stability properties, and classification under certain geometric decay conditions.

Contribution

It proves that higher-dimensional catenoids have index one and classifies $rac 2n$-stable minimal hypersurfaces as either catenoids or hyperplanes under decay conditions.

Findings

Higher dimensional catenoids have index one.

Catenoids are $rac 2n$-stable minimal hypersurfaces.

Complete $rac 2n$-stable minimal hypersurfaces are either catenoids or hyperplanes.

Abstract

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.