A Curvature Gap for Minimal Submanifolds in Spheres

Let $F:M^n\to {\text{Sn}}^{n+q}(1)$ be a closed connected minimal immersion in the unit sphere with second fundamental form $h$, $n\ge 3$, $q\ge 2$, and $S=\mathrm{\mid }h{\mathrm{\mid }}^2$. We prove that if $M$ is not totally geodesic, then $$\underset{M}{\max }S\ge \frac{2n}{3}+\frac{n-2}{6300(39n+8)}\ge \frac{2n}{3}+\frac{1}{787500}\mathrm{.}$$