On the index of minimal hypersurfaces of spheres

Let $M\subset S^{n+1}\subset {\mathbb{R}}^{n+2}$ be a compact minimal hypersurface of the $n$-dimensional Euclidean unit sphere. Let us denote by $\mathrm{\mid }A{\mathrm{\mid }}^2$ the square of the norm of the second fundamental form and $J(f)=-\Delta f-nf-|A|^2f$ the stability operator. It is known that the index (the number of negative eigenvalues of $J$) is 1 when $M$ is a totally geodesic sphere, and it is $n+3$ when $M$ is a Clifford minimal hypersurface. It has been conjectured that for any other minimal hypersurface, the index must be greater than $n+3$. One partial result for this conjecture states that if the index is $n+3$ and $M$ is not Clifford, then ${\int }_M\mathrm{\mid }A{\mathrm{\mid }}^2<n\mathrm{\mid }M\mathrm{\mid }$ where $\mathrm{\mid }M\mathrm{\mid }$ is the $n$ dimensional volume of $M$. Somehow this partial result states that if the index of $M$ is $n+3$ then the average of the function $\mathrm{\mid }A{\mathrm{\mid }}^2$ needs to be small. In this note we prove that this average cannot be very small. We will show that for any pair of positive numbers ${\delta }_1$ and ${\delta }_2$ with ${\delta }_1+{\delta }_2=1$, if ${\int }_M\mathrm{\mid }A{\mathrm{\mid }}^2\le {\delta }_{2}n\mathrm{\mid }M\mathrm{\mid }$ and $\mathrm{\mid }A{\mathrm{\mid }}^2(x)\le 2n{\delta }_1$ for all $x\in M$, then the index of $M$ is greater than $n+3$.