First stability eigenvalue characterization of Clifford hypersurfaces

Abstract:

The stability operator of a compact oriented minimal hypersurface $M^{n-1}\subset S^n$ is given by $J = -\Delta -\|A\|^2-(n-1)$, where $\|A\|$ 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 $\beta \ge 0$ 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 $|M|$ denote the area of $M$ and let $g$ denote the genus of $M$. We will prove that $\beta |M|\ge 8\pi (g-1)$. 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 $|M|\le \frac {16 \pi }{1-\beta }$.