A Characterization of Quadric Constant Mean Curvature Hypersurfaces of Spheres |
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Authors: | Luis J Alías Jr" target="_blank">Aldir BrasilJr Oscar Perdomo |
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Institution: | (1) Departamento de Matemáticas, Universidad de Murcia, Campus de Espinardo, 30100 Espinardo, Murcia, Spain;(2) Departamento de Matemática, Universidade Federal do Ceará, Campus do Pici, 60455-760 Fortaleza, Ceará, Brazil;(3) Department of Mathematical Sciences, Central Connecticut State University, New Britain, CT 06050, USA |
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Abstract: | Let
be an immersion of a complete n-dimensional oriented manifold. For any v∈ℝ
n+2, let us denote by ℓ
v
:M→ℝ the function given by ℓ
v
(x)=〈φ(x),v〉 and by f
v
:M→ℝ, the function given by f
v
(x)=〈ν(x),v〉, where
is a Gauss map. We will prove that if M has constant mean curvature, and, for some v≠0 and some real number λ, we have that ℓ
v
=λ
f
v
, then, φ(M) is either a totally umbilical sphere or a Clifford hypersurface. As an application, we will use this result to prove that
the weak stability index of any compact constant mean curvature hypersurface M
n
in
which is neither totally umbilical nor a Clifford hypersurface and has constant scalar curvature is greater than or equal
to 2n+4.
A. Brasil Jr. was partially supported by CNPq, Brazil, 306626/2007-1. |
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Keywords: | Constant mean curvature Clifford hypersurface Stability operator First eigenvalue |
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