Triharmonic hypersurfaces with constant mean curvature in pseudo-Riemannian space forms

Abstract

In this paper, triharmonic hypersurfaces with constant mean curvature in pseudo-Riemannian space forms are studied. Under the assumption that the shape operator is diagonalizable, we first classify completely the nonminimal hypersurfaces with at most two distinct principal curvatures and give some examples of non-biharmonic triharmonic hypersurfaces. Then, we prove that the hypersurfaces with at most four distinct principal curvatures have constant scalar curvature. As a consequence, we obtain that such triharmonic hypersurfaces in pseudo-Euclidean spaces are minimal, which gives an affirmative partial solution to the generalized Chen's conjecture in [21].