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 $M^m⟶({\mathbb{R}}^{m+1},{\delta }_{ij})$ in a Euclidean space becomes a biharmonic hypersurface $M^m⟶({\mathbb{R}}^{m+1},f^{-2}{\delta }_{ij})$ in the conformally flat space. Our examples include all biharmonic hypersurfaces found in [Ou1] and [OT] as special cases.

16 pages

References in corpus (1)

• Surfaces with parallel mean curvature in $\mathbb{C}{P}^n\times \mathbb{R}$ and $\mathbb{C}{H}^n\times \mathbb{R}$