The matter of biharmonic surfaces of the 3-dimensional Euclidean space has been studied (firstly) from a differential geometric point of view by Bang-Yen Chen and others, who has showed that the only biharmonic surfaces in E3 are minimal ones. In general, the biharmonicity condition on any hypersurface x : Mn → En+1 is defined by Δ2x = 0, where Δ is the Laplace operator on Mn. Many people have paid attention to various extensions of Chen’s theorem. In this paper, we approve an advanced version of the theorem, replacing Δ by the operator L1, which stands for the linearized operator of the first variation of the 2-th mean curvature arising from the normal variations of Mn in En+1. In the case n = 4, for any L1-biharmonic hypersurface x : M4 → E5, having assumed that it has three distinct principal curvatures and constant ordinary mean curvature, we prove that, M4 has to be 1-minimal.
