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Proceedings of the American Mathematical Society

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Compact graphs over a sphere of constant second order mean curvature


Authors: A. Barros and P. Sousa
Journal: Proc. Amer. Math. Soc. 137 (2009), 3105-3114
MSC (2000): Primary 53C42, 53C45; Secondary 53C65
DOI: https://doi.org/10.1090/S0002-9939-09-09862-1
Published electronically: April 23, 2009
MathSciNet review: 2506469
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Abstract: The aim of this work is to show that a compact smooth star-shaped hypersurface $ \Sigma^n$ in the Euclidean sphere $ \mathbb{S}^{n+1}$ whose second function of curvature $ S_2$ is a positive constant must be a geodesic sphere $ \mathbb{S}^{n}(\rho)$. This generalizes a result obtained by Jellett in $ 1853$ for surfaces $ \Sigma^2$ with constant mean curvature in the Euclidean space $ \mathbb{R}^3$ as well as a recent result of the authors for this type of hypersurface in the Euclidean sphere $ \mathbb{S}^{n+1}$ with constant mean curvature. In order to prove our theorem we shall present a formula for the operator $ L_{r}(g)=div\left(P_r\nabla g\right)$ associated with a new support function $ g$ defined over a hypersurface $ M^n$ in a Riemannian space form $ M_{c}^{n+1}$.


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Additional Information

A. Barros
Affiliation: Departamento de Matemática, Universidade Federal do Ceará, 60455-760 Fortaleza, Brazil
Email: abbarros@mat.ufc.br

P. Sousa
Affiliation: Departamento de Matemática, Universidade Federal do Piauí, 64049-550 Teresina, Brazil
Email: pauloalexandre@ufpi.edu.br

DOI: https://doi.org/10.1090/S0002-9939-09-09862-1
Keywords: $L_r$ operator, radial graph, constant scalar curvature
Received by editor(s): August 7, 2007
Received by editor(s) in revised form: December 27, 2008
Published electronically: April 23, 2009
Additional Notes: The first author was partially supported by CNPq-BR
The second author was partially supported by CAPES-BR
Communicated by: Richard A. Wentworth
Article copyright: © Copyright 2009 American Mathematical Society