Polyharmonic hypersurfaces into pseudo-Riemannian space forms

In this paper we shall assume that the ambient manifold is a pseudo-Riemannian space form $N^{m+1}_t(c)$ of dimension $m+1$ and index $t$ ($m\geq2$ and $1 \leq t\leq m$). We shall study hypersurfaces $M^{m}_{t'}$ which are polyharmonic of order $r$ (briefly, $r$-harmonic), where $r\geq 3$ and either $t'=t$ or $t'=t-1$. Let $A$ denote the shape operator of $M^{m}_{t'}$. Under the assumptions that $M^{m}_{t'}$ is CMC and $Tr A^2$ is a constant, we shall obtain the general condition which determines that $M^{m}_{t'}$ is $r$-harmonic. As a first application, we shall deduce the existence of several new families of proper $r$-harmonic hypersurfaces with diagonalizable shape operator, and we shall also obtain some results in the direction that our examples are the only possible ones provided that certain assumptions on the principal curvatures hold. Next, we focus on the study of isoparametric hypersurfaces whose shape operator is non-diagonalizable and also in this context we shall prove the existence of some new examples of proper $r$-harmonic hypersurfaces ($r \geq 3$). Finally, we shall obtain the complete classification of proper $r$-harmonic isoparametric pseudo-Riemannian surfaces into a $3$-dimensional Lorentz space form.