Biharmonic submanifolds in a Riemannian manifold with non-positive   curvature
  We show that for an isometric immersion of a complete Riemannian manifold into a Riemannian manifold with non-positive curvature, the norm of the mean curvature vector field is square integrable, then it is minimal. This is a partial affirmative answer of the B. Y. Chen's conjecture. 
Nobumitsu Nakauchi and Hajime Urakawa
math.DG