On the index of minimal hypersurfaces of spheres

Abstract

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 $|A|^2$ 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 $\int_M |A|^2<n|M|$ where $|M|$ 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 $|A|^2$ 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 $\int_M |A|^2\le \delta_2 n|M|$ and $|A|^2(x)\le2n\delta_1$ for all $x\in M$, then the index of $M$ is greater than $n+3$.