Generalized Chern conjecture for constant-mean-curvature hypersurfaces

Let $M$ be a closed hypersurface with constant mean curvature and constant scalar curvature in the unit sphere $\mathbb{S}^{n + 1}$. Generalized Chern conjecture. For each $n$ and $H$, the set of all possible values of $S$ is discrete, where $H$ is the mean curvature and $S$ is the squared norm of the second fundamental form. This extends Chern's conjecture from minimal hypersurfaces to constant-mean-curvature hypersurfaces. The stated discreteness problem remains open in general; the paper presents gap theorems and records several partial results for special dimensions or additional hypotheses.