We consider a complete biharmonic immersed submanifold M in a Euclidean space E N . Assume that the immersion is proper, that is, the preimage of every compact set in E N is also compact in M . Then, we prove that M is minimal. It is considered as an affirmative answer to the global version of Chen's conjecture for biharmonic submanifolds.

J. Eells and L. Lemaire introduced k-harmonic maps, and Shaobo Wang showed the first variational formula. When k = 2, it is called biharmonic maps (2-harmonic maps). There have been extensive studies in the area. In this paper, we consider the relationship between biharmonic maps and k-harmonic maps, and we show the non-existence theorem of 3-harmonic maps. We also give the definition of k-harmonic submanifolds of Euclidean spaces and study k-harmonic curves in Euclidean spaces. Furthermore, we give a conjecture for k-harmonic submanifolds of Euclidean spaces.

A triharmonic map is a critical point of the 3-energy in the space of smooth maps between two Riemannian manifold. We study the generalized Chen's conjecture for a triharmonic isometric immersion ϕ into a space form of non-positively constant curvature. We show that if the domain is complete and both the 4-energy of ϕ, and the L 4 -norm of the tension field τ (ϕ), are finite, then such an immersion ϕ is minimal.2000 Mathematics Subject Classification. primary 58E20, secondary 53C43.

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Abstract. We consider a complete nonnegative biminimal submanifold M (that is, a complete biminimal submanifold with λ ≥ 0) in a Euclidean space E N . Assume that the immersion is proper, that is, the preimage of every compact set in E N is also compact in M . Then, we prove that M is minimal. From this result, we give an affirmative partial answer to Chen's conjecture. For the case of λ < 0, we construct examples of biminimal submanifolds and curves.

We consider biharmonic maps φ : (M, g) → (N, h) from a complete Riemannian manifold into a Riemannian manifold with non-positive sectional curvature. Assume that α satisfies 1 < α < ∞. If for such an α, M |τ (φ)| α dvg < ∞ and M |dφ| 2 dvg < ∞, where τ (φ) is the tension field of φ, then we show that φ is harmonic. For a biharmonic submanifold, we obtain that the above assumption M |dφ| 2 dvg < ∞ is not necessary. These results give affirmative partial answers to the global version of generalized Chen's conjecture.