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 Rm by parallel hyperplanes into a foliation with each leave a proper biharmonic hypersurface. We also characterize proper biharmonic vertical cylinders in S × R and H × 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 (φ,Ω) = 1 2 ∫ Ω |τ(φ)| dx for every compact subset Ω of M , where τ(φ) = Traceg∇dφ is the tension field of φ. The Euler-Lagrange equation of this functional gives the biharmonic map equation ([15]) (1) τ (φ) := Traceg(∇∇ −∇φ∇M )τ(φ)− TracegR (dφ, τ(φ))dφ = 0, Date: 01/08/2009. 1991 Mathematics Subject Classification. 58E20, 53C12, 53C42.