Triharmonic CMC hypersurfaces in space forms with 4 distinct principal curvatures

Abstract

A triharmonic map is a critical point of the tri-energy in the space of smooth maps between two Riemannian manifolds. In this paper, we prove that if $M^n(n\ge 4)$ is a CMC proper triharmonic hypersurface in a space form ${\mathbb{R}}^{n+1}(c)$ with four distinct principal curvatures and the multiplicity of the zero principal curvature is at most one, then $M$ has constant scalar curvature. In particular, we obtain any CMC proper triharmonic hypersurface in ${\mathbb{R}}^5(c)$ is minimal when $c\le 0$, which supports the generalized Chen's conjecture. We also give some characterizations of CMC proper triharmonic hypersurfaces in ${\mathbb{S}}^5$.

11 pages