Let \(x:M^n\rightarrow E^m\) be an isometric immersion into Euclidean space. According to the well known Beltrami’s formula, we have \(\Delta x = H\), where \(\Delta\) is the Laplacian of \((M,x)\) and \(H\) its mean curvature vector, so that Euclidean minimal submanifolds are characterized as those with harmonic position vector. B.-Y. Chen [Soochow J. Math. 17, 169-188 (1991; Zbl 0749.53037)]posed the problem of classifying Euclidean submanifolds with harmonic mean curvature vector, that is, those submanifolds satisfying \(\Delta^2x=0\), which are therefore called biharmonic submanifolds. He also conjectured that the only biharmonic Euclidean submanifolds are the minimal ones. The conjecture is known to be true in some special cases: surfaces in \(E^3\) (B.-Y. Chen), curves and submanifolds with constant mean curvature (I. Dimitric), and hypersurfaces in \(E^4\) (T. Hasanis and T. Vlachos), for instance.
In this paper, the author gives a new proof of the Hasanis-Vlachos result (who gave the result in a more ample context) by using a method which is coordinate independent and does not appeal to the use of the computer, in the hope that it can be easier generalized to higher dimensional hypersurfaces.