First stability eigenvalue characterization of Clifford hypersurfaces

The stability operator of a compact oriented minimal hypersurface $M^{n-1}\subset S^n$ is given by $J = -\Delta -\Vert A\Vert^2-(n-1)$, where $\Vert A\Vert$ is the norm of the second fundamental form. Let $\lambda_1$ be the first eigenvalue of $J$ and define $\beta = -\lambda_1-2(n-1)$. In 1968 Simons proved that $\beta\ge0$ for any non-equatorial minimal hypersurface $M\subset S^n$. In this paper we will show that $\beta=0$ only for Clifford hypersurfaces. For minimal surfaces in $S^3$, let $\vert M\vert$ denote the area of $M$ and let $g$ denote the genus of $M$. We will prove that $\beta\vert M\vert\ge 8\pi(g-1)$. Moreover, if $M$is embedded, then we will prove that $\beta \ge \frac{g-1}{g+1}$. If in addition to the embeddeness condition we have that $\beta<1$, then we will prove that $\vert M\vert\le \frac{16 \pi}{1-\beta}$.