On the stability index of hypersurfaces with constant mean curvature in spheres
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- by Luis J. Alías, Aldir Brasil Jr. and Oscar Perdomo PDF
- Proc. Amer. Math. Soc. 135 (2007), 3685-3693 Request permission
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 $\mathbb {S}^{n+1}$. In this paper we prove that the weak index of any other compact constant mean curvature hypersurface $M^n$ in $\mathbb {S}{n+1}$ 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 $\sqrt {k/(n+2)}\leqslant r\leqslant \sqrt {(k+2)/(n+2)}$.References
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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
- 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
- © Copyright 2007 American Mathematical Society
- 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
- MathSciNet review: 2336585