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 $n⩾3$ and $q⩾2$. 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 \[ S_{\max}>\frac{2n}{3}+\frac{n-2}{182}. \]

arXiv comment33 pages