A Curvature Gap for Minimal Submanifolds in Spheres

Abstract

Let $F:M^n\to\Sn^{n+q}(1)$ be a closed minimal immersion in the unit sphere, with $n\ge3$, $q\ge2$. Let $S$ be the squared length of its second fundamental form. We prove that if $M$ is not totally geodesic, then $$ \max_M S>\frac{n(288\sqrt6\,n-288)}{(288\sqrt6+192)n-137}>\frac{2n}{3}+\frac{31-9\sqrt6}{75}(n-2). $$