Chern Conjecture on Minimal Willmore Hypersurfaces with Constant Scalar Curvature

Abstract: In this paper, we prove that for an $n$-dimensional closed minimal Willmore hypersurface $M<sup>n$ with constant scalar curvature in the unit sphere $\mathbb{S}<sup>{n+1}$, the squared norm $S$ of the second fundamental form of $M<sup>n$ satisfies $$S\geqslant n+\frac{4n+9-\sqrt{4 n^{2}+60 n+81}}{2}$$ if $S&gt;n$. This proves, in the approximate sense, the Chern conjecture about the second gap ($S\geqslant 2n$ if $S&gt;n$), which will be fully verified under a further inequality condition about the 4-th mean curvature.