Compact graphs over a sphere of constant second order mean curvature

Abstract:

The aim of this work is to show that a compact smooth star-shaped hypersurface $\Sigma ^n$ in the Euclidean sphere $\mathbb {S}^{n+1}$ whose second function of curvature $S_2$ is a positive constant must be a geodesic sphere $\mathbb {S}^{n}(\rho )$. This generalizes a result obtained by Jellett in $1853$ for surfaces $\Sigma ^2$ with constant mean curvature in the Euclidean space $\mathbb {R}^3$ as well as a recent result of the authors for this type of hypersurface in the Euclidean sphere $\mathbb {S}^{n+1}$ with constant mean curvature. In order to prove our theorem we shall present a formula for the operator $L_{r}(g)=div\left (P_r\nabla g\right )$ associated with a new support function $g$ defined over a hypersurface $M^n$ in a Riemannian space form $M_{c}^{n+1}$.