A new characterization of r -stable hypersurfaces in space forms

H. F. de Lima; M. A. Velásquez

Archivum Mathematicum (2011)

  • Volume: 047, Issue: 2, page 119-131
  • ISSN: 0044-8753

Abstract

top
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.

How to cite

top

de Lima, H. F., and Velásquez, M. A.. "A new characterization of $r$-stable hypersurfaces in space forms." Archivum Mathematicum 047.2 (2011): 119-131. <http://eudml.org/doc/116540>.

@article{deLima2011,
abstract = {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.},
author = {de Lima, H. F., Velásquez, M. A.},
journal = {Archivum Mathematicum},
keywords = {space forms; $r$-th mean curvatures; $r$-stability; space form; -th mean curvature; -stability},
language = {eng},
number = {2},
pages = {119-131},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {A new characterization of $r$-stable hypersurfaces in space forms},
url = {http://eudml.org/doc/116540},
volume = {047},
year = {2011},
}

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
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 047
IS - 2
SP - 119
EP - 131
AB - 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.
LA - eng
KW - space forms; $r$-th mean curvatures; $r$-stability; space form; -th mean curvature; -stability
UR - http://eudml.org/doc/116540
ER -

References

top
  1. Alencar, H., do Carmo, M., Colares, A. G., 10.1007/BF03025712, Math. Z. 213 (1993), 117–131. (1993) Zbl0792.53057MR1217674DOI10.1007/BF03025712
  2. Barbosa, J. L. M., Colares, A. G., 10.1023/A:1006514303828, Ann. Global Anal. Geom. 15 (1997), 277–297. (1997) Zbl0891.53044MR1456513DOI10.1023/A:1006514303828
  3. Barbosa, J. L. M., do Carmo, M., 10.1007/BF01215045, Math. Z. 185 (3) (1984), 339–353. (1984) Zbl0513.53002MR0731682DOI10.1007/BF01215045
  4. Barbosa, J. L. M., do Carmo, M., Eschenburg, J., 10.1007/BF01161634, Math. Z. 197 (1) (1988), 123–138. (1988) Zbl0653.53045MR0917854DOI10.1007/BF01161634
  5. Barros, A., Sousa, P., 10.1090/S0002-9939-09-09862-1, Proc. Amer. Math. Soc. 137 (2009), 3105–3114. (2009) Zbl1189.53058MR2506469DOI10.1090/S0002-9939-09-09862-1
  6. Caminha, A., 10.1016/j.geomphys.2005.06.007, J. Geom. Phys. 56 (2006), 1144–1174. (2006) Zbl1102.53044MR2234043DOI10.1016/j.geomphys.2005.06.007
  7. Cheng, S. Y., Yau, S. T., 10.1007/BF01425237, Math. Ann. 225 (1977), 195–204. (1977) Zbl0349.53041MR0431043DOI10.1007/BF01425237
  8. de Lima, H. F., 10.1016/j.geomphys.2006.07.005, J. Geom. Phys. 57 (2007), 967–975. (2007) Zbl1111.53049MR2275203DOI10.1016/j.geomphys.2006.07.005
  9. He, Y., Li, H., 10.1007/s10455-007-9095-3, Ann. Global Anal. Geom. 34 (2008), 55–68. (2008) Zbl1156.53037MR2415178DOI10.1007/s10455-007-9095-3
  10. Reilly, R., Variational properties of functions of the mean curvatures for hypersurfaces in space forms, J. Differential Equations 8 (1973), 465–477. (1973) Zbl0277.53030MR0341351
  11. Rosenberg, H., Hypersurfaces of constant curvature in space forms, Bull. Sci. Math. 117 (1993), 217–239. (1993) Zbl0787.53046MR1216008
  12. Xin, Y., Minimal submanifolds and related topics, World Scientific Publishing co., Singapore, 2003. (2003) Zbl1055.53047MR2035469

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.