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


Authors: Luen-fai Tam and Detang Zhou
Journal: Proc. Amer. Math. Soc. 137 (2009), 3451-3461
MSC (2000): Primary 53A10; Secondary 53C42
DOI: https://doi.org/10.1090/S0002-9939-09-09962-6
Published electronically: May 7, 2009
MathSciNet review: 2515414
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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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Additional Information

Luen-fai Tam
Affiliation: The Institute of Mathematical Sciences and Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong, People’s Republic of China
Email: lftam@math.cuhk.edu.hk

Detang Zhou
Affiliation: Instituto de Matematica, Universidade Federal Fluminense, Centro, Niterói, RJ 24020-140, Brazil
Email: zhou@impa.br

DOI: https://doi.org/10.1090/S0002-9939-09-09962-6
Keywords: Catenoid, minimal hypersurface, stability.
Received by editor(s): January 26, 2009
Published electronically: May 7, 2009
Additional Notes: The first author’s research was partially supported by Earmarked Grant of Hong Kong #CUHK403005
The second author’s research was supported by CNPq and FAPERJ of Brazil.
Communicated by: Chuu-Lian Terng
Article copyright: © Copyright 2009 American Mathematical Society

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