Abstract: In this paper, we study L² harmonic 2-forms on a complete hypersurface M of Euclidean space Rⁿ⁺¹(n ≥ 3). By applying the Bochner technique, we prove that if the Lⁿ(M) norms of the traceless second fundamental form Φ and the mean curvature vector H are both bounded from above by certain constants which depend only on n, then the L² harmonic 2-forms on M are parallel. Furthermore, if M is a non-minimal hypersurface, then there is no nontrivial L² harmonic 2-form on M.

Key words: Euclidean space, hypersurface, L² harmonic 2-forms, non-minimal