Lk-BIHARMONIC HYPERSURFACES IN THE EUCLIDEAN SPACE

Chen conjecture states that every Euclidean biharmonic submanifold is minimal. In this paper we consider the Chen conjecture for L-k-operators. The new conjecture (L-k-conjecture) is formulated as follows: If L(k)(2)x = 0 then Hk+1 = 0 where x : M-n -> Rn+1 is an isometric immersion of a Riemannian manifold M-n into the Euclidean space Rn+1, Hk+1 is the (k+1)-th mean curvature of M, and L-k is the linearized operator of the (k + 1)-th mean curvature of the Euclidean hypersurface M. We prove the L-k-conjecture for the hypersurface M with at most two principal curvatures.