Nonexistence of minimal Lagrangian spheres in hyperKähler manifolds

Abstract.

We prove that for $n>1$ one cannot immerse $S^{2n}$ as a minimal Lagrangian manifold into a hyperKähler manifold. More generally we show that any minimal Lagrangian immersion of an orientable closed manifold $L^{2n}$ into a hyperKähler manifold $H^{4n}$ must have nonvanishing second Betti number $\beta_2$ and that if $\beta_2=1$, $L^{2n}$ is a Kähler manifold and more precisely a Kähler 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.