A lower bound for the norm of the second fundamental form of minimal hypersurfaces of $\mathbb{S}^{n+1}$

Abstract.

The aim of this paper is to give an estimate for the squared norm S of the second fundamental form A of a compact minimal hypersurface $M^{n} \subset \mathbb{S}^{n+1}$ in terms of the gap $n - \lambda_{1}$ , where $\lambda_{1}$ stands for the first eigenvalue of the Laplacian of M. More precisely we will show that there exists a constant $k \geq \frac{n}{n-1}$ such that $S \geq k \frac {n-1}{n} (n - \lambda_{1})$ .