Conformal-biharmonic hypersurfaces in spheres and product spaces

Abstract 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 isoparametric 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 hypersurfaces 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 non-constant mean curvature, we establish that a totally umbilical c -biharmonic hypersurface must necessarily be totally geodesic.