Polyharmonic hypersurfaces into space forms

In this paper we shall assume that the ambient manifold is a space form $N^{m+1}(c)$ and we shall consider polyharmonic hypersurfaces of order $r$ (briefly, $r$-harmonic), where $r\geq 3$ is an integer. For this class of hypersurfaces we shall prove that, if $c \leq 0$, then any $r$-harmonic hypersurface must be minimal provided that the mean curvature function and the squared norm of the shape operator are constant. When the ambient space is $\mathbb{S}^{m+1}$, we shall obtain the geometric condition which characterizes the $r$-harmonic hypersurfaces with constant mean curvature and constant squared norm of the shape operator, and we shall establish the bounds for these two constants. In particular, we shall prove the existence of several new examples of proper $r$-harmonic isoparametric hypersurfaces in $\mathbb{S}^{m+1}$ for suitable values of $m$ and $r$. Finally, we shall show that all these $r$-harmonic hypersurfaces are also $ES-r$-harmonic, i.e., critical points of the Eells-Sampson $r$-energy functional.