A characterization of Clifford tori with constant scalar curvature one by the first stability eigenvalue

Abstract.

Let M be a compact hypersurface with constant scalar curvature one immersed into the unit Euclidean sphere $\mathbb{S}^{{n + 1}}$. As is well-known, such hypersurfaces can be characterized variationally as critical points of the integral ∫ _M Hdv. In this paper we derive a sharp upper bound for the first eigenvalue of the corresponding Jacobi operator in terms of the mean curvature of the hypersurface. Moreover, we prove that this bound is achieved only for the Clifford tori in $\mathbb{S}^{{n + 1}}$ with scalar curvature one.