Kobayashi pseudometric on hyperkahler manifolds

The Kobayashi pseudometric on a complex manifold is the maximal pseudometric such that any holomorphic map from the Poincaŕe disk to the manifold is distance-decreasing. Kobayashi has conjectured that this pseudometric vanishes on Calabi-Yau manifolds. Using ergodicity of complex structures, we prove this for all hyperkähler manifold with b₂ ≥ 7 that admits a deformation with a Lagrangian fibration and whose Picard rank is not maximal. The Strominger- Yau-Zaslow (SYZ) conjecture claims that parabolic nef line bundles on hyperkähler manifolds are semi-ample. We prove that the Kobayashi pseudometric vanishes for any hyperkähler manifold with b₂ ≥ 7 if the SYZ conjecture holds for all its deformations. This proves the Kobayashi conjecture for all K3 surfaces and their Hilbert schemes.