A k-harmonic map is a critical point of the k-energy defined on space smooth maps between two Riemannian manifolds. In this paper, we prove that if $$M^{n} (n\ge 3)$$ CMC proper triharmonic hypersurface with at most three distinct principal curvatures in form $$\mathbb {R}^{n+1}(c)$$ , then M has constant scalar curvature. This supports generalized Chen’s conjecture when $$c\le 0$$ . When $$c=1...
