Hypersurfaces of E-4 with harmonic mean curvature vector

A submanifold M-n of a Euclidean space E-m is said to have harmonic mean cu rvature vector field if Delta (H) over right arrow = (0) over right arrow, where (H) over right arrow denotes the mean curvature vector. B.-Y. CHEN co njectured that the only submanifolds of Euclidean spaces with harmonic mean curvature vector field, are the minimal ones. In this paper, we give a pro of of the theorem that every hypersurface of E-4 with harmonic mean curvatu re vector field is minimal. The method gives insight in the role of the pri ncipal curvatures.