A spectral characterization of the $H(r)$-torus by the first stability eigenvalue
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- by Luis J. Alías, Abdênago Barros and Aldir Brasil Jr. PDF
- Proc. Amer. Math. Soc. 133 (2005), 875-884 Request permission
Abstract:
Let $M$ be a compact hypersurface with constant mean curvature immersed into the unit Euclidean sphere $\mathbb {S}^{n+1}$. In this paper we derive a sharp upper bound for the first eigenvalue of the stability operator of $M$ in terms of the mean curvature and the length of the total umbilicity tensor of the hypersurface. Moreover, we prove that this bound is achieved only for the so-called $H(r)$-tori in $\mathbb {S}^{n+1}$, with $r^2\leq (n-1)/n$. This extends to the case of constant mean curvature hypersurfaces previous results given by Wu (1993) and Perdomo (2002) for minimal hypersurfaces.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
- Abdênago Barros
- Affiliation: Departamento de Matemática, Universidade Federal do Ceará, Campus do Pici, 60455-760 Fortaleza-Ce, Brazil
- Email: abbarros@mat.ufc.br
- Aldir Brasil Jr.
- Affiliation: Departamento de Matemática, Universidade Federal do Ceará, Campus do Pici, 60455-760 Fortaleza-Ce, Brazil
- Email: aldir@mat.ufc.br
- Received by editor(s): August 26, 2003
- Received by editor(s) in revised form: October 27, 2003
- Published electronically: September 16, 2004
- Additional Notes: The first author was partially supported by DGCYT, BFM2001-2871, MCYT, and Fundación Séneca, PI-3/00854/FS/01, Spain.
The second author was partially supported by FINEP, Brazil
The third author was partially supported by CAPES, BEX0324/02-7, Brazil - Communicated by: Jon G. Wolfson
- © Copyright 2004 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 133 (2005), 875-884
- MSC (2000): Primary 53C42; Secondary 53A10
- DOI: https://doi.org/10.1090/S0002-9939-04-07559-8
- MathSciNet review: 2113939
Dedicated: Dedicated to Professor J. Lucas Barbosa on the occasion of his 60th birthday