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

Abstract

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 combines a symmetrized Hessian estimate, an integral identity for a mixed moment, and a weighted estimate for the repeated-index components of the covariant derivative of the second fundamental form. An exact polynomial argument treats the entire excluded interval. We also prove an explicit positive lower bound for the average of the normalized quadratic defect $P/S^2$, where $P=f_4-f_3^2/S-S^2/n$. This second estimate holds in every dimension $n\geq2$ and gives a quantitative obstruction to concentration near the locus of two principal curvatures.