Biminimal properly immersed submanifolds in the Euclidean spaces

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