Nonexistence of minimal Lagrangian spheres in hyperKahler manifolds

We prove chat for n > 1 one cannot immerse S-2n as a minimal Lagrangian manifold into a hyperKahler manifold. More generally we show that any minimal Lagrangian immersion of an orientable closed manifold L-2n into a hyperKahler manifold H-4n must have nonvanishing second Betti number beta(2) and that if beta(2) = 1, L-2n is a Kahler manifold and more precisely a Kahler submanifold in H-4n w.r.t. one of the complex structures on H-4n. In addition we derive a result for the other Betti numbers.