Let x: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 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 δ(2)-ideal and δ(3)-ideal hypersurfaces of a Euclidean space of arbitrary dimension. © 2012 Elsevier B.V.
Chen, B. Y., & Munteanu, M. I. (2013). Biharmonic ideal hypersurfaces in Euclidean spaces. Differential Geometry and Its Application, 31(1), 1–16. https://doi.org/10.1016/j.difgeo.2012.10.008