Stable Hypersurfaces in a 4-Dimensional Sphere

Abstract:

      We study complete noncompact $1$-minimal stable hypersurfaces in a $4$-dimensional sphere $\mathbb{S}^{4}$. We show that there is no complete noncompact $1$-minimal stable hypersurfaces in $\mathbb{S}^{4}$ with polynomial volume growth and the restriction of the mean curvature and Gauss-Kronecker curvature. These results are partial answers to the conjecture of Alencar, do Carmo and Elbert when the ambient space is a $4$-dimensional sphere.