Polyharmonic hypersurfaces into complex space forms

We characterize homogeneous hypersurfaces in complex space forms which arise as critical points of a higher order energy functional. As a consequence, we obtain existence and non-existence results for \(\mathbb{C}\mathbb{P}^n\) $\mathbb{C}{\mathbb{P}}^n$ and \(\mathbb{C}\mathbb{H}^n\) $\mathbb{C}{\mathbb{H}}^n$ , respectively. Moreover, we study the stability of biharmonic hypersurfaces and compute the normal index for a large family of solutions.