Weak Kobayashi conjecture for Calabi–Yau manifolds

Establish whether Calabi–Yau manifolds are non-Kobayashi-hyperbolic, equivalently whether every Calabi–Yau manifold contains a nonconstant entire curve, as predicted by the weak form of the Kobayashi conjecture.

Background

Kobayashi hyperbolicity means that a compact complex manifold contains no nonconstant entire curve. The paper notes that Kähler hyperbolicity implies Kobayashi hyperbolicity and proves that the fundamental group of a Calabi–Yau manifold cannot coincide with that of a Kähler hyperbolic manifold. This result is presented as partial evidence toward the unresolved weak form of the Kobayashi conjecture for Calabi–Yau manifolds.