Biharmonic ideal hypersurfaces in Euclidean spaces

Let $x:M\to {\mathbb{E}}^m$ be an isometric immersion from a Riemannian n-manifold into a Euclidean m-space. Denote by Δ and $\overrightarrow{x}$ the Laplace operator and the position vector of M, respectively. Then M is called biharmonic if ${\mathrm{\Delta }}^2\overrightarrow{x}=0$. The following Chen?s 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 $\delta (2)$-ideal and $\delta (3)$-ideal hypersurfaces of a Euclidean space of arbitrary dimension.