Estimate for index of closed minimal hypersurfaces in spheres

Abstract

The aim of this work is to deal with index of closed orientable non-totally geodesic minimal hypersurface Σⁿ of the Euclidean unit sphere Sⁿ⁺¹ whose second fundamental form has squared norm bounded from below by n. In this case we shall show that the index of stability, denoted by Ind_Σ, is greater than or equal to n + 3, with equality occurring at only Clifford tori $\mathbf{S}^k(\frac{k}{n})\times\mathbf{S}^{n-k}(\sqrt{\frac{(n-k)}{n}})$. Moreover, we shall prove also that, besides Clifford tori, we have the following gap: Ind_Σ ≥ 2n + 5.