Local rigidity of constant mean curvature hypersurfaces in space forms (II)

This is the second article of a sequence of research on the local rigidity of constant mean curvature (CMC) hypersurfaces in space forms. In the previous one, we studied the local rigidity of CMC hypersurfaces whose the number of the distinct principal curvatures satisfies $g\le 3$. In this paper, we study the local rigidity of CMC hypersurfaces with $g\ge 4$. When $g>4$, we prove that if the $k$-order mean curvatures $H_k$, $(k=2,\cdots ,g-1)$ are constant and there exist enough multiple principal curvatures, then the CMC hypersurface is an isoparametric hypersurface. When $g=4$, if $H_2$ and $H_3$ are constant, then the CMC hypersurface is an isoparametric hypersurface.