Chern Conjecture on Minimal Willmore Hypersurfaces with Constant Scalar Curvature

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