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

Published electronically:
June 22, 2007

MathSciNet review:
2336585

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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 .

**1.**Lucas Barbosa and Pierre Bérard,*Eigenvalue and “twisted” eigenvalue problems, applications to CMC surfaces*, J. Math. Pures Appl. (9)**79**(2000), no. 5, 427–450 (English, with English and French summaries). MR**1759435**, 10.1016/S0021-7824(00)00160-4**2.**J. Lucas Barbosa, Manfredo do Carmo, and Jost Eschenburg,*Stability of hypersurfaces of constant mean curvature in Riemannian manifolds*, Math. Z.**197**(1988), no. 1, 123–138. MR**917854**, 10.1007/BF01161634**3.**Marcel Berger, Paul Gauduchon, and Edmond Mazet,*Le spectre d’une variété riemannienne*, Lecture Notes in Mathematics, Vol. 194, Springer-Verlag, Berlin-New York, 1971 (French). MR**0282313****4.**T. E. Cecil and P. J. Ryan,*Tight and taut immersions of manifolds*, Research Notes in Mathematics, vol. 107, Pitman (Advanced Publishing Program), Boston, MA, 1985. MR**781126****5.**Ahmad El Soufi,*Applications harmoniques, immersions minimales et transformations conformes de la sphère*, Compositio Math.**85**(1993), no. 3, 281–298 (French). MR**1214448****6.**I. Guadalupe, Aldir Brasil Jr., and J. A. Delgado,*A characterization of the Clifford torus*, Rend. Circ. Mat. Palermo (2)**48**(1999), no. 3, 537–540. MR**1731453**, 10.1007/BF02844342**7.**Oscar Perdomo,*Low index minimal hypersurfaces of spheres*, Asian J. Math.**5**(2001), no. 4, 741–749. MR**1913819**, 10.4310/AJM.2001.v5.n4.a8**8.**James Simons,*Minimal varieties in riemannian manifolds*, Ann. of Math. (2)**88**(1968), 62–105. MR**0233295****9.**Francisco Urbano,*Minimal surfaces with low index in the three-dimensional sphere*, Proc. Amer. Math. Soc.**108**(1990), no. 4, 989–992. MR**1007516**, 10.1090/S0002-9939-1990-1007516-1

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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:
http://dx.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