A Curvature Gap for Minimal Submanifolds in Spheres

Abstract

Let $F:M^n\to\Sn^{n+q}(1)$ be a closed connected minimal immersion in the unit sphere with second fundamental form $h$, $n\ge3$, $q\ge2$, and $S=|h|^2$. We prove that if $M$ is not totally geodesic, then \[ \max_M S\ge \frac{2n}{3}+\frac{n-2}{6300(39n+8)} \ge\frac{2n}{3}+\frac1{787500}. \]