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A spectral characterization of the $H(r)$-torus by the first stability eigenvalue


Authors: Luis J. Alías, Abdênago Barros and Aldir Brasil Jr.
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
Published electronically: September 16, 2004
MathSciNet review: 2113939
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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.


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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

DOI: https://doi.org/10.1090/S0002-9939-04-07559-8
Keywords: Constant mean curvature, $H(r)$-torus, stability operator, first eigenvalue
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
Dedicated: Dedicated to Professor J. Lucas Barbosa on the occasion of his 60th birthday
Communicated by: Jon G. Wolfson
Article copyright: © Copyright 2004 American Mathematical Society

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