Conformal-biharmonic hypersurfaces in spheres and product spaces

The conformal-bienergy functional E₂ᶜ is a modified version of the classical bienergy functional E₂ and it is conformally invariant in the case of a four-dimensional domain. The critical points of E₂ᶜ are called conformal-biharmonic and denoted c -biharmonic. In the first part of the paper we study the c -biharmonic hypersurfaces Mᵐ with constant principal curvatures in the product space Lᵐ(ε ) × R , where Lᵐ(ε ) denotes a space form of constant sectional curvature ε . Specifically, we demonstrate that Mᵐ is either totally geodesic or a cylindrical hypersurface of the form Mᵐ⁻¹ × R , where Mᵐ⁻¹ is an isoparametric c -biharmonic hypersurface in Lᵐ(ε ) . In the second part of this article we obtain a full description of isoparametric c -biharmonic hypersurfaces in Sᵐ⁺¹ and a complete classification of c -biharmonic hypersurfaces with constant scalar curvature in Sᵐ⁺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 S⁵ . In the final part of the paper, as a preliminary effort to understand c -biharmonic hypersurfaces in Lᵐ(ε ) × R with non-constant mean curvature, we establish that a totally umbilical c -biharmonic hypersurface must necessarily be totally geodesic.