Triharmonic CMC Hypersurfaces in Space Forms with at Most 3 Distinct Principal Curvatures

Abstract

A k-harmonic map is a critical point of the k-energy defined on the space of smooth maps between two Riemannian manifolds. In this paper, we prove that if $$M^{n} (n\ge 3)$$ is a CMC proper triharmonic hypersurface with at most three distinct principal curvatures in a space form $$\mathbb {R}^{n+1}(c)$$ , then M has constant scalar curvature. This supports the generalized Chen’s conjecture when $$c\le 0$$ . When $$c=1$$ , we give an optimal upper bound of the mean curvature H for a non-totally umbilical proper CMC k-harmonic hypersurface with constant scalar curvature in a sphere. As an application, we give the complete classification of the 3-dimensional complete proper CMC triharmonic hypersurfaces in $$\mathbb {S}^{4}$$ .