On biminimal submanifolds in nonpositively curved manifolds

Biminimal immersions are critical points of the bienergy for normal variations with fixed energy, that is critical points of the functional E2(⋅)+λE(⋅), λ∈R, for normal variations. A submanifold is called a biminimal submanifold if it is a biminimal isometric immersion. In this note we prove that positive (that is λ>0) complete biminimal submanifolds in nonpositively curved manifolds are minimal. Furthermore for nonpositive (that is λ≤0) complete biminimal submanifolds in negative space forms we get this result under certain proper conditions. Our result is sharp by the examples from [14]. These results inspire us to make two conjectures on biminimal submanifolds.