Biharmonic hypersurfaces in a conformally flat space

Abstract

Biharmonic hypersurfaces in a generic conformally flat space are studied in this paper. The equation of such hypersurfaces is derived and is used to determine the conformally flat metric $f^{-2}\delta_{ij}$ on the Euclidean space $\mathbb{R}^{m+1}$ so that a minimal hypersurface $M^m\longrightarrow (\mathbb{R}^{m+1}, \delta_{ij})$ in a Euclidean space becomes a biharmonic hypersurface $M^m\longrightarrow (\mathbb{R}^{m+1}, f^{-2}\delta_{ij})$ in the conformally flat space. Our examples include all biharmonic hypersurfaces found in [Ou1] and [OT] as special cases.