Hypersurfaces of \(\bar\mathbb{E}^ 4\) satisfying \(\Delta\vec H=\lambda\vec H\) (Q1371329)

Let \(M^n\) be an \(n\)-dimensional connected submanifold of the Euclidean space \(E^m\). Denote by \(H\) and \(\Delta\) the mean curvature vector field of \(M^n\) and the Laplace operator with respect to the induced metric, respectively. The study of the submanifolds satisfying (*) \(\Delta H =\lambda H\), \(\lambda \in \mathbb{R}\), was initiated by B.-Y. Chen in 1988, and arose in the context of his theory of submanifolds of finite type. In the present paper the author proves the following: A hypersurface of \(E^4\) satisfying (*) must have constant mean curvature. The same result, but by another method, has been proved by \textit{Th. Hasanis} and \textit{Th. Vlachos} [Ann. Global Anal. Geom. 13, 69-77 (1995; Zbl 0823.53045)].