The pinching constant for closed minimal submanifolds of high codimension in the sphere

Abstract

Let $M^n$ be a closed minimal submanifold in the unit sphere $\mathbb{S}^{n+q}$ with $n\geqslant 3$ and $q\geqslant 2$. Let $S$ be the squared length of its second fundamental form and $S_{\max}=\max_{p\in M}S(p)$. We prove that if $M$ is not totally geodesic, then \[ S_{\max}>\frac{2n}{3}+\frac{n-2}{182}. \]