On triharmonic hypersurfaces in space forms

In this paper we study triharmonic hypersurfaces immersed in a space form $N^{n+1}(c)$. We prove that any proper CMC triharmonic hypersurface in the sphere $\mathbb S^{n+1}$ has constant scalar curvature; any CMC triharmonic hypersurface in the hyperbolic space $\mathbb H^{n+1}$ is minimal. Moreover, we show that any CMC triharmonic hypersurface in the Euclidean space $\mathbb R^{n+1}$ is minimal provided that the multiplicity of the principal curvature zero is at most one. In particular, we are able to prove that every CMC triharmonic hypersurface in the Euclidean space $\mathbb R^{6}$ is minimal.These results extend some recent works due to Montaldo-Oniciuc-Ratto and Chen-Guan, and give affirmative answer to the generalized Chen's conjecture.