A note on conformal-biharmonic hypersurfaces

Abstract: The conformal bienergy functional $E_2<sup>c$ was recently introduced as a modified version of the classical bienergy functional $E_2$ in order to ensure the validity of some conformal invariance properties. The critical points of $E_2<sup>c$ are called conformal-biharmonic and denoted $c$-biharmonic. In this paper we study the $c$-biharmonic hypersurfaces $M<sup>m$ with constant principal curvatures in the product space $${\mathbb L}^{m(\varepsilon)} \times \mathbb{R}$$, where ${\mathbb L}^{m(\varepsilon)}$ denotes a space form of constant sectional curvature $\varepsilon$. Specifically, we demonstrate that $M^m$ is either totally geodesic or a cylindrical hypersurface of the form $M^{m-1} \times \mathbb{R}$, where $M^{m-1}$ is a $c$-biharmonic isoparametric hypersurface in ${\mathbb L}^{m(\varepsilon)}$. To provide further insight, we describe the structure of $c$-biharmonic isoparametric hypersurfaces in space forms. In the final part, as a preliminary effort to understand $c$-biharmonic hypersurfaces $M^m$ in $${\mathbb L}^{m(\varepsilon)} \times \mathbb{R}$$ with non constant mean curvature, we establish that a totally umbilical $c$-biharmonic hypersurface must necessarily be totally geodesic.