Kobayashi pseudometric on hyperkähler manifolds

The Kobayashi pseudometric on a complex manifold M is the maximal pseudometric such that any holomorphic map from the Poincaré disk to M is distance-decreasing. Kobayashi has conjectured that this pseudometric vanishes on Calabi-Yau manifolds. Using ergodicity of complex structures, we prove this result for any hyperkähler manifold if it admits a deformation with a Lagrangian fibration, and its Picard rank is not maximal. The SYZ conjecture claims that any parabolic nef line bundle on a deformation of a given hyperkähler manifold is semi-ample. We prove that the Kobayashi pseudometric vanishes for all hyperkähler manifolds satisfying the SYZ property. This proves the Kobayashi conjecture for K3 surfaces and their Hilbert schemes.