Biharmonic hypersurfaces in Riemannian manifolds

Abstract

We study biharmonic hypersurfaces in a generic Riemannian manifold. We first derive an invariant equation for such hypersurfaces generalizing the biharmonic hypersurface equation in space forms studied in \cite{Ji2}, \cite{CH}, \cite{CMO1}, \cite{CMO2}. We then apply the equation to show that the generalized Chen's conjecture is true for totally umbilical biharmonic hypersurfaces in an Einstein space, and construct a (2-parameter) family of conformally flat metrics and a (4-parameter) family of multiply warped product metrics each of which turns the foliation of an upper-half space of $\mathhbb{R}^m$ by parallel hyperplanes into a foliation with each leave a proper biharmonic hypersurface. We also characterize proper biharmonic vertical cylinders in $S^2\times \mathbb{R}$ and $H^2\times \mathbb{R}$.