Biharmonic submanifolds in a Riemannian manifold
  In this paper, we solve affirmatively B.-Y. Chen's conjecture for hypersurfaces in the Euclidean space, under a generic condition. More precisely, every biharmonic hypersurface of the Euclidean space must be minimal if their principal curvatures are simple, and the associated frame field is irreducible. 
N. Koiso, H. Urakawa
math.DG