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
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
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 in n+1 which is not totally umbilical and has constant scalar curvature is greater than or equal to
, with equality if and only if
is a constant mean curvature Clifford torus
with radius
.
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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