Stable hypersurfaces in spheres with constant scalar curvature

We study complete noncompact 1-minimal stable hypersurfaces M in Sn+1. We obtain that there is no stable complete noncompact 1-minimal hypersurface in S-4 with nonzero Gauss-Kronecker curvature and finite total curvature. We show that there is no stable complete noncompact 1-minimal hypersurface in S 4 with polynomial volume growth, nonzero Gauss-Kronecker curvature and sup(M) H-2 < 5/2. At last, we discuss stable complete noncompact hypersurfaces M with constant scalar curvature R > 1 in Sn+1 and show another nonexistence result. These results are generalised versions of results of Alencar, Santos, Zhou and Silva Neto on stable hypersurfaces in Rn+1 with constant scalar curvature.