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 IndΣ, 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: IndΣ ≥ 2n + 5.