Compact graphs over a sphere of constant second order mean curvature
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- by A. Barros and P. Sousa PDF
- Proc. Amer. Math. Soc. 137 (2009), 3105-3114 Request permission
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 $M_{c}^{n+1}$.References
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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
- 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
- © Copyright 2009 American Mathematical Society
- 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
- MathSciNet review: 2506469