An improved second pinching theorem for minimal hypersurfaces with constant scalar curvature in spheres

Let $M^n$ be a closed minimal hypersurface of the unit sphere with constant squared norm $S$ of the second fundamental form. For $n\geq4$, we prove that $S>n$ implies $S>85n/59$, improving the bound $10n/7$ of Suh and Yang. No constancy assumption is imposed on the cubic trace $f_3$. The proof combi