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On the Gauss map of hypersurfaces with constant scalar curvature in spheres

Authors: Hilário Alencar, Harold Rosenberg and Walcy Santos
Journal: Proc. Amer. Math. Soc. 132 (2004), 3731-3739
MSC (2000): Primary 53C40; Secondary 53A10
Published electronically: July 12, 2004
MathSciNet review: 2084098
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Abstract: In this work we consider connected, complete and orientable hypersurfaces of the sphere $\mathbb{S} ^{n+1}$ with constant nonnegative $r$-mean curvature. We prove that under subsidiary conditions, if the Gauss image of $M$ is contained in a closed hemisphere, then $M$ is totally umbilic.

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Additional Information

Hilário Alencar
Affiliation: Departamento de Matemática, Universidade Federal de Alagoas, 57072-900, Maceió, AL, Brazil

Harold Rosenberg
Affiliation: Institut de Mathématiques de Jussieu, 2 Place Jussieu, 75251 Paris, France

Walcy Santos
Affiliation: Departamento de Matemática, Universidade Federal do Rio de Janeiro, Caixa Postal 68530, 21945-970 Rio de Janeiro, RJ, Brazil

Keywords: $r$-mean curvature, spheres, Gauss image
Received by editor(s): April 28, 2003
Received by editor(s) in revised form: September 1, 2003
Published electronically: July 12, 2004
Additional Notes: The first and third authors’ research was partially supported by CNPq and the French-Brazilian Agreement in Mathematics
Communicated by: Richard A. Wentworth
Article copyright: © Copyright 2004 American Mathematical Society