Geodesic
A new characterization of $r$-stable hypersurfaces in space forms
Archivum mathematicum, Tome 47 (2011) no. 2, pp. 119-131 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

In this paper we study the $r$-stability of closed hypersurfaces with constant $r$-th mean curvature in Riemannian manifolds of constant sectional curvature. In this setting, we obtain a characterization of the $r$-stable ones through of the analysis of the first eigenvalue of an operator naturally attached to the $r$-th mean curvature.
In this paper we study the $r$-stability of closed hypersurfaces with constant $r$-th mean curvature in Riemannian manifolds of constant sectional curvature. In this setting, we obtain a characterization of the $r$-stable ones through of the analysis of the first eigenvalue of an operator naturally attached to the $r$-th mean curvature.
Classification : 53B30, 53C42, 53C50, 53Z05, 83C99
Keywords: space forms; $r$-th mean curvatures; $r$-stability 
@article{ARM_2011_47_2_a5,
     author = {de Lima, H. F. and Vel\'asquez, M. A.},
     title = {A new characterization of $r$-stable hypersurfaces in space forms},
     journal = {Archivum mathematicum},
     pages = {119--131},
     year = {2011},
     volume = {47},
     number = {2},
     mrnumber = {2813538},
     zbl = {1249.53081},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ARM_2011_47_2_a5/}
}
TY  - JOUR
AU  - de Lima, H. F.
AU  - Velásquez, M. A.
TI  - A new characterization of $r$-stable hypersurfaces in space forms
JO  - Archivum mathematicum
PY  - 2011
SP  - 119
EP  - 131
VL  - 47
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/ARM_2011_47_2_a5/
LA  - en
ID  - ARM_2011_47_2_a5
ER  - 
%0 Journal Article
%A de Lima, H. F.
%A Velásquez, M. A.
%T A new characterization of $r$-stable hypersurfaces in space forms
%J Archivum mathematicum
%D 2011
%P 119-131
%V 47
%N 2
%U http://geodesic.mathdoc.fr/item/ARM_2011_47_2_a5/
%G en
%F ARM_2011_47_2_a5
de Lima, H. F.; Velásquez, M. A. A new characterization of $r$-stable hypersurfaces in space forms. Archivum mathematicum, Tome 47 (2011) no. 2, pp. 119-131. http://geodesic.mathdoc.fr/item/ARM_2011_47_2_a5/

[1] Alencar, H., do Carmo, M., Colares, A. G.: Stable hypersurfaces with constant scalar curvature. Math. Z. 213 (1993), 117–131. | DOI | MR | Zbl

[2] Barbosa, J. L. M., Colares, A. G.: Stability of hypersurfaces with constant $r$-mean curvature. Ann. Global Anal. Geom. 15 (1997), 277–297. | DOI | MR | Zbl

[3] Barbosa, J. L. M., do Carmo, M.: Stability of hypersurfaces with constant mean curvature. Math. Z. 185 (3) (1984), 339–353. | DOI | MR | Zbl

[4] Barbosa, J. L. M., do Carmo, M., Eschenburg, J.: Stability of hypersurfaces of constant mean curvature in Riemannian manifolds. Math. Z. 197 (1) (1988), 123–138. | DOI | MR | Zbl

[5] Barros, A., Sousa, P.: Compact graphs over a sphere of constant second order mean curvature. Proc. Amer. Math. Soc. 137 (2009), 3105–3114. | DOI | MR | Zbl

[6] Caminha, A.: On spacelike hypersurfaces of constant sectional curvature Lorentz manifolds. J. Geom. Phys. 56 (2006), 1144–1174. | DOI | MR | Zbl

[7] Cheng, S. Y., Yau, S. T.: Hypersurfaces with constant scalar curvature. Math. Ann. 225 (1977), 195–204. | DOI | MR | Zbl

[8] de Lima, H. F.: Spacelike hypersurfaces with constant higher order mean curvature in de Sitter space. J. Geom. Phys. 57 (2007), 967–975. | DOI | MR | Zbl

[9] He, Y., Li, H.: Stability of area-preserving variations in space forms. Ann. Global Anal. Geom. 34 (2008), 55–68. | DOI | MR | Zbl

[10] Reilly, R.: Variational properties of functions of the mean curvatures for hypersurfaces in space forms. J. Differential Equations 8 (1973), 465–477. | MR | Zbl

[11] Rosenberg, H.: Hypersurfaces of constant curvature in space forms. Bull. Sci. Math. 117 (1993), 217–239. | MR | Zbl

[12] Xin, Y.: Minimal submanifolds and related topics. World Scientific Publishing co., Singapore, 2003. | MR | Zbl