On the Gauss map of hypersurfaces with constant scalar curvature in spheres
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- by Hilário Alencar, Harold Rosenberg and Walcy Santos PDF
- Proc. Amer. Math. Soc. 132 (2004), 3731-3739 Request permission
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.References
- H. Alencar, M. do Carmo, and A. G. Colares, Stable hypersurfaces with constant scalar curvature, Math. Z. 213 (1993), no. 1, 117–131. MR 1217674, DOI 10.1007/BF03025712
- João Lucas Marques Barbosa and Antônio Gervasio Colares, Stability of hypersurfaces with constant $r$-mean curvature, Ann. Global Anal. Geom. 15 (1997), no. 3, 277–297. MR 1456513, DOI 10.1023/A:1006514303828
- J. L. Barbosa and M. do Carmo, Stable minimal surfaces, Bull. Amer. Math. Soc. 80 (1974), 581–583. MR 336533, DOI 10.1090/S0002-9904-1974-13508-1
- Bernstein, S. - Sur un théorème de Géométrie et ses applications aux équations aux dérivées partialles du type elliptique. Comm. de la Soc. Math. de Kharkov (2ième sér.) 15, 38–45 (1915/1917).
- P. S. Bullen, D. S. Mitrinović, and P. M. Vasić, Means and their inequalities, Mathematics and its Applications (East European Series), vol. 31, D. Reidel Publishing Co., Dordrecht, 1988. Translated and revised from the Serbo-Croatian. MR 947142, DOI 10.1007/978-94-017-2226-1
- Ennio De Giorgi, Una estensione del teorema di Bernstein, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 19 (1965), 79–85 (Italian). MR 178385
- G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1988. Reprint of the 1952 edition. MR 944909
- Jorge Hounie and Maria Luiza Leite, Two-ended hypersurfaces with zero scalar curvature, Indiana Univ. Math. J. 48 (1999), no. 3, 867–882. MR 1736975, DOI 10.1512/iumj.1999.48.1664
- Katsumi Nomizu and Brian Smyth, On the Gauss mapping for hypersurfaces of constant mean curvature in the sphere, Comment. Math. Helv. 44 (1969), 484–490. MR 257939, DOI 10.1007/BF02564548
- Robert C. Reilly, Extrinsic rigidity theorems for compact submanifolds of the sphere, J. Differential Geometry 4 (1970), 487–497. MR 290296
- Robert C. Reilly, Variational properties of functions of the mean curvatures for hypersurfaces in space forms, J. Differential Geometry 8 (1973), 465–477. MR 341351
- Harold Rosenberg, Hypersurfaces of constant curvature in space forms, Bull. Sci. Math. 117 (1993), no. 2, 211–239. MR 1216008
- James Simons, Minimal varieties in riemannian manifolds, Ann. of Math. (2) 88 (1968), 62–105. MR 233295, DOI 10.2307/1970556
Additional Information
- Hilário Alencar
- Affiliation: Departamento de Matemática, Universidade Federal de Alagoas, 57072-900, Maceió, AL, Brazil
- Email: hilario@mat.ufal.br
- Harold Rosenberg
- Affiliation: Institut de Mathématiques de Jussieu, 2 Place Jussieu, 75251 Paris, France
- MR Author ID: 150570
- Email: rosen@math.jussieu.fr
- Walcy Santos
- Affiliation: Departamento de Matemática, Universidade Federal do Rio de Janeiro, Caixa Postal 68530, 21945-970 Rio de Janeiro, RJ, Brazil
- Email: walcy@im.ufrj.br
- 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
- © Copyright 2004 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 132 (2004), 3731-3739
- MSC (2000): Primary 53C40; Secondary 53A10
- DOI: https://doi.org/10.1090/S0002-9939-04-07493-3
- MathSciNet review: 2084098