Stability properties for the higher dimensional catenoid in ℝⁿ⁺¹

Tam, Luen-fai; Zhou, Detang

doi:10.1090/S0002-9939-09-09962-6

Stability properties for the higher dimensional catenoid in $\mathbb R^{n+1}$
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by Luen-fai Tam and Detang Zhou PDF

Proc. Amer. Math. Soc. 137 (2009), 3451-3461 Request permission

Abstract:

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.

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