A spectral characterization of the -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 | References | Similar Articles | Additional Information

Abstract: Let be a compact hypersurface with constant mean curvature immersed into the unit Euclidean sphere . In this paper we derive a sharp upper bound for the first eigenvalue of the stability operator of 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 -tori in , with . This extends to the case of constant mean curvature hypersurfaces previous results given by Wu (1993) and Perdomo (2002) for minimal hypersurfaces.

**1.**H. Alencar and M. do Carmo, Hypersurfaces with constant mean curvature in spheres,*Proc. Amer. Math. Soc.***120**(1994), 1223-1229. MR**94f:53108****2.**J.L. Barbosa and P. Bérard, Eigenvalue and ``twisted" eigenvalue problems. Applications to CMC surfaces,*J. Math. Pures Appl. (9)***79**(2000), 427-450.MR**2001f:58064****3.**J.L. Barbosa, M. do Carmo and J. Eschenburg, Stability of hypersurfaces with constant mean curvature in Riemannian manifolds,*Math. Z.***197**(1988), 123-138.MR**88m:53109****4.**Abdênago Alves de Barros, Aldir Chaves Brasil Jr., and Luis Amancio Machado de Sousa Jr.,*A new characterization of submanifolds with parallel mean curvature vector in 𝑆^{𝑛+𝑝}*, Kodai Math. J.**27**(2004), no. 1, 45–56. MR**2042790**, https://doi.org/10.2996/kmj/1085143788**5.**S.S. Chern, M. do Carmo, and S. Kobayashi, Minimal submanifolds of a sphere with second fundamental form of constant length. 1970*Functional Analysis and Related Fields*(Proc. Conf. for M. Stone, Univ. Chicago, Chicago, Ill., 1968) Springer, New York, pp. 59-75. MR**42:8424****6.**A. El Soufi and S. Ilias, Majoration de la seconde valeur propre d'un opérateur de Schrödinger sur une variété compacte et applications,*J. Funct. Anal.***103**(1992), 294-316. MR**93g:58150****7.**A. El Soufi and S. Ilias, Second eigenvalue of Schrödinger operators and mean curvature,*Comm. Math. Phys.***208**(2000), 761-770. MR**2001g:58050****8.**H.B. Lawson Jr., Local rigidity theorems for minimal hypersurfaces,*Ann. of Math. (2)***89**, 187-197. MR**38:6505****9.**M. Okumura, Hypersurfaces and a pinching problem on the second fundamental tensor,*Amer. J. Math.***96**(1974), 207-213. MR**50:5701****10.**O. Perdomo, First stability eigenvalue characterization of Clifford hypersurfaces,*Proc. Amer. Math. Soc.***130**(2002), 3379-3384.MR**2003f:53109****11.**J. Simons, Minimal varieties in Riemannian manifolds,*Ann. of Math. (2)*,**88**(1968), 62-105. MR**38:1617****12.**C. Wu, New characterizations of the Clifford tori and the Veronese surface,*Arch. Math. (Basel)*,**61**(1993), 277-284.MR**94h:53084**

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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