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 $n\ge 3$, $q\ge 2$. Let $S$ be the squared length of its second fundamental form. We prove that if $M$ is not totally geodesic, then $$\underset{M}{\max }S>\frac{n(288\sqrt{6}n-288)}{(288\sqrt{6}+192)n-137}>\frac{2n}{3}+\frac{31-9\sqrt{6}}{75}(n-2).$$

24 pages. Using AI to provide better inequality estimates and improve constants