Dense entire curves in rationally connected manifolds

We show the existence of metrically dense entire curves (of growth order 0) in rationally connected complex projective manifolds, confirming for this case a conjecture formulated by the first-named author, according to which such entire curves on projective manifolds exist if and only if these are special in the sense of birational geometry. We next show the existence of dense entire curves, avoiding the singular locus, in certain log-terminal normal rational surfaces. This implies via results of Grassi and Oguiso the existence of dense entire curves into any Calabi–Yau threefold fibered in abelian surfaces or elliptic curves. We then show that a dense entire curve may be chosen on any rationally connected manifold in such a way that it does not lift to a given ramified cover, answering in this case a question of Corvaja and Zannier about the Nevanlinna analog of the ‘weak Hilbert property’ of arithmetic geometry. As shown in the appendix by Kollár, this entire curve may be chosen independently of the ramified cover.