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 Σ n in the Euclidean sphere S n+1 whose second function of curvature S 2 is a positive constant must be a geodesic sphere S n (ρ). This generalizes a result obtained by Jellett in 1853 for surfaces Σ 2 with constant mean curvature in the Euclidean space ℝ 3 as well as a recent result of the authors for this type of hypersurface in the Euclidean sphere 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(P r ∇g) associated with a new support function g defined over a hypersurface M n in a Riemannian space form M n+1 c .