On biharmonic conformal hypersurfaces

Abstract: In this paper, we first derive biharmonic equation for conformal hypersurfaces in a generic Riemannian manifold generalizing that for biharmonic hypersurfaces in \cite{Ou1} and that for biharmonic conformal surfaces in \cite{Ou3, Ou2, Ou4}. We then show that if a totally umbilical hypersurface in a space form admits a biharmonic conformal immersion into the ambient space, then the conformal factor has to be an isoparametric function. We also prove that no part of a non-minimal totally umbilical hypersurface in a space form of nonpositive curvature admits a biharmonic conformally immersion into that space form whilst, for the positive curvature space form, we show that the totally umbilical hypersurface $$S^{4(\frac{\sqrt{3}}{2})\hookrightarrow} S<sup>5$$ does admit a biharmonic conformal immersion into $S<sup>5$.