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Conformal-biharmonic hypersurfaces in spheres and product spaces

The conformal-bienergy functional $E_2^c$ is a modified version of the classical bienergy functional $E_2$ and it is conformally invariant in the case of a four-dimensional domain. The critical points of $E_2^c$ are called conformal-biharmonic and denoted $c$-biharmonic. In the first part of the paper we study the $c$-biharmonic hypersurfaces $M^m$ 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 an iso\-parametric $c$-biharmonic hypersurface in $ {\mathbb L}^m(\varepsilon) $. In the second part of this article we obtain a full description of isoparametric $c$-biharmonic hypersurfaces in $\mathbb{S}^{m+1}$ and a complete classification of $c$-biharmonic hypersurfaces with constant scalar curvature in $\mathbb{S}^{m+1}$, $m=2,3$ and $m=4$ with an additional assumption. In this context, we shall also prove a global result for compact $c$-biharmonic immersions in $\mathbb{S}^5$. In the final part of the paper, as a preliminary effort to understand $c$-biharmonic hypersurfaces in $ {\mathbb L}^m(\varepsilon) \times \mathbb{R} $ with \textit{non-constant} mean curvature, we establish that a totally umbilical $c$-biharmonic hypersurface must necessarily be totally geodesic.