Local rigidity of constant mean curvature hypersurfaces in space forms

In this paper, we study the local rigidity of constant mean curvature (CMC) hypersurfaces. Let $x:M^n\to M^{n+1}\left(c\right),n\ge 4$, be a piece of immersed constant mean curvature hypersurface in the $(n+1)$-dimensional space form $M^{n+1}\left(c\right)$. We prove that if the scalar curvature R is constant and the number g of the distinct principal curvatures satisfies $g\le 3$, then $M^n$ is an isoparametric hypersurface. Further, if $M^n$ is a minimal hypersurface, then $M^n$ is a totally geodesic hypersurface for $c\le 0$, and $M^n$ is either a Cartan minimal hypersurface, a Clifford minimal hypersurface, or a totally geodesic hypersurface for $c>0$, which solves the high dimensional version of Bryant Conjecture.