L1-Biharmonic Hypersurfaces in Euclidean Spaces with Three Distinct Principal Curvatures

A submanifold Mn of the Euclidean space En+m is said to be biharmonic if its position map x : Mn ! En+m satises the condition 2x = 0, where stands for the Laplace operator. A well-known conjecture of Bang-Yen Chen says that, the only biharmonic submanifolds of Euclidean spaces are the minimal ones. In this paper, we consider a modied version of the conjecture, replacing by its extension, L1-operator (namely, L1-conjecture). The L1-conjecture states that any L1-biharmonic Euclidean hypersurface is 1-minimal. We prove that the L1-conjecture is true for L1-biharmonic hypersurfaces with three distinct principal curvatures and constant mean curvature of a Euclidean space of arbitrary dimension.