Biconservative hypersurfaces in Euclidean Spaces

Abstract

An isometric immersion $X:{\Sigma }^n⟶{\mathbb{E}}^{n+1}$ is biharmonic (BHH) if ${\Delta }^{2}X=0$, i.e., if $\Delta H=0$, where $\Delta$ and $H$ are the metric Laplacian and the mean curvature vector field of ${\Sigma }^n$ respectively. More generally, biconservative hypersurfaces (BCHs) are isometric immersions for which only the tangential part of the biharmonic equation vanishes. BCHs are first described and then constructed.