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 𝕊^n+1 has constant scalar curvature; any CMC triharmonic hypersurface in the hyperbolic space ℍ^n+1 is minimal. Moreover, we show that any CMC triharmonic hypersurface in the space ℝ^n+1 is minimal provided that zero is a principal curvature of multiplicity at most one. In particular, we are able to prove that every CMC triharmonic hypersurface in the Euclidean space 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.