The Kobayashi pseudometric $d_M$ on a complex manifold $M$ is the maximal pseudometric such that any holomorphic map from the Poincare disk to $M$ is distance-decreasing. Kobayashi conjectured that this pseudometric vanishes on Calabi-Yau manifolds, and in particular, Calabi-Yau manifolds have "entire curves". Using ergodicity of complex structures, together with S. Lu and M. Verbitsky we prove this conjecture for all K3 surfaces and for many classes of hyperk\"ahler manifolds. We define the Kobayashi quotient of a complex variety by identifying points with vanishing Kobayashi pseudodistance between them. Together with F. Bogomolov, S. Lu and M. Verbitsky we prove that the Kobayashi quotients associated to ergodic complex structures on a compact manifold are isomorphic.