Biharmonic hypersurfaces in Riemannian manifolds

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 [16], [8], [6], [7]. 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 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 × R and H 2 × R. 1. Biharmonic maps and submanifolds All manifolds, maps, and tensor fields appear in this paper are supposed to be smooth unless there is an otherwise statement. A biharmonic map is a map ϕ: (M, g) − → (N, h) between Riemannian manifolds that is a critical point of the bienergy functional E 2 (ϕ, Ω) = 1