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 in 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 closed proper CMC triharmonic hypersurfaces in ${\mathbb{S}}^4$.

15 pages; Corollary 1.9 is added and Theorem 1.10 is slightly modified; some references are update