ON THE STABILITY OF SPACELIKE HYPERSURFACES WITH HIGHER ORDER MEAN CURVATURE IN A DE SITTER SPACE

Abstract

Abstract. The closed spacelike hypersurfaces with higher order meancurvature is discussed in a de Sitter space. The hypersurface is provedstable if and only if it is totally umbilical. 1. IntroductionAs we all know, the hypersurfaces with constant mean curvature or con-stant scalar curvature in real space forms are characterized as critical points ofthe area functional for volume-preserving variations. Many results have beenachieved about hypersurfaces with constant mean curvature or constant scalarcurvature in a unit sphere S n+1 (1) [1, 2, 3]. Among these results, the geodesicsphere is the only stable compact hypersurface with constant mean curvaturein a sphere as in [3]. After that, the closed hypersurfaces with higher ordermean curvature immersed in a Riemannian space form are studied and similarresults are obtained by other researches [4, 8, 12].Achievements are not only obtained in Riemannian space, in fact, many re-searches are also conducted in Lorentzian spaces. Constant mean curvaturespacelike hypersurfaces are solutions to a variational problems. Actually, theyare the critical points of the area functional for variations that leave constanta certain volume function. In this sense, Barbosa and Oliker [5] computed thesecond variation formula and obtained in the de Sitter space S