A Curvature Gap for Minimal Submanifolds in Spheres

Abstract: Let $F:M<sup>n\to\Sn<sup>{n+q}(1)$ be a closed connected minimal immersion in the unit sphere with $n\ge3$, $q\ge2$, and $S=|h|<sup>2$. We prove that if $M$ is not totally geodesic, then [ \max_M S\ge \frac{2n}{3}+2^{-2n^2-10n-30}n^{-2n-2}. ]