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 \( S \geq k \frac {n-1}{n} (n - \lambda_{1}) \) .