Biharmonic hypersurfaces with three distinct principal curvatures in Euclidean 5-space

A submanifold MnMn of a Euclidean space EmEm is said to be biharmonic if ΔH⃗=0, where ΔΔ is a rough Laplacian operator and H⃗ denotes the mean curvature vector. In 1991, B.Y. Chen proposed a well-known conjecture that the only biharmonic submanifolds of Euclidean spaces are the minimal ones. In this paper, we prove that Chen’s conjecture is true for the case of hypersurfaces with three distinct principal curvatures in Euclidean 5-spaces.