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:Mⁿ→Mⁿ⁺¹(c),n≥4, be a piece of immersed constant mean curvature hypersurface in the (n+1)-dimensional space form Mⁿ⁺¹(c). We prove that if the scalar curvature R is constant and the number g of the distinct principal curvatures satisfies g≤3, then Mⁿ is an isoparametric hypersurface. Further, if Mⁿ is a minimal hypersurface, then Mⁿ is a totally geodesic hypersurface for c≤0, and Mⁿ 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.