Biharmonic Hypersurfaces in Euclidean Spaces

An isometric immersion $X:{\mathrm{\Sigma }}^n⟶{\mathbb{E}}^{n+1}$ is biharmonic if ${\mathrm{\Delta }}^{2}X=0$, i.e. if $\mathrm{\Delta }H=0$, where $\mathrm{\Delta }$ and $H$ are the metric Laplacian and the mean curvature vector field of ${\mathrm{\Sigma }}^n$ respectively. More generally, biconservative hypersurfaces (BCH) are isometric immersions for which only the tangential part of the biharmonic equation vanishes. We study and construct BCH that are holonomic, i.e. for which the principal curvature directions define an integrable net, and we deduce that ${\mathrm{\Sigma }}^n$ is a holonomic biharmonic hypersurface iff it is minimal.