Biharmonic ideal hypersurfaces in Euclidean spaces

Let x:M→Emx:M→Em be an isometric immersion from a Riemannian n-manifold into a Euclidean m  -space. Denote by Δ and x→ the Laplace operator and the position vector of M, respectively. Then M   is called biharmonic if Δ2x→=0. The following Chenʼs Biharmonic Conjecture made in 1991 is well-known and stays open: The only biharmonic submanifolds of Euclidean spaces are the minimal ones  . In this paper we prove that the biharmonic conjecture is true for δ(2)δ(2)-ideal and δ(3)δ(3)-ideal hypersurfaces of a Euclidean space of arbitrary dimension.