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On the stability index of hypersurfaces with constant mean curvature in spheres


Authors: Luis J. Alías, Aldir Brasil Jr. and Oscar Perdomo
Journal: Proc. Amer. Math. Soc. 135 (2007), 3685-3693
MSC (2000): Primary 53C42; Secondary 53A10
DOI: https://doi.org/10.1090/S0002-9939-07-08886-7
Published electronically: June 22, 2007
MathSciNet review: 2336585
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Abstract: Barbosa, do Carmo and Eschenburg characterized the totally umbilical spheres as the only weakly stable compact constant mean curvature hypersurfaces in the Euclidean sphere . In this paper we prove that the weak index of any other compact constant mean curvature hypersurface $ M^n$ in n+1 which is not totally umbilical and has constant scalar curvature is greater than or equal to $ n+2$, with equality if and only if $ M$ is a constant mean curvature Clifford torus $ \mathbb{S}^{k}(r)\times\mathbb{S}^{n-k}(\sqrt{1-r^2})$ with radius $ \sqrt{k/(n+2)}\leqslant r\leqslant\sqrt{(k+2)/(n+2)}$.


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Additional Information

Luis J. Alías
Affiliation: Departamento de Matemáticas, Universidad de Murcia, Campus de Espinardo, E-30100 Espinardo, Murcia, Spain
Email: ljalias@um.es

Aldir Brasil Jr.
Affiliation: Departamento de Matemática, Universidade Federal do Ceará, Campus do Pici, 60455-760 Fortaleza-Ce, Brazil
Email: aldir@mat.ufc.br

Oscar Perdomo
Affiliation: Departamento de Matemáticas, Universidad del Valle, Cali, Colombia
Email: osperdom@mafalda.univalle.edu.co

DOI: https://doi.org/10.1090/S0002-9939-07-08886-7
Keywords: Constant mean curvature, $H(r)$-torus, stability operator, first eigenvalue
Received by editor(s): August 2, 2005
Received by editor(s) in revised form: August 11, 2006
Published electronically: June 22, 2007
Additional Notes: The first author was partially supported by MEC/FEDER project MTM2004-04934-C04-02, Spain, and by the Fundación Séneca project 00625/PI/04, Spain
The second author was partially supported by CNPq, Brazil
The third author was partially supported by Colciencias, Colombia
Communicated by: Richard A. Wentworth
Article copyright: © Copyright 2007 American Mathematical Society

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