Orders of meromorphic mappings into Hopf and Inoue surfaces

Abstract

In a late paper of J. Noguchi and J. Winkelmann [7] (J. Math. Soc. Jpn., Vol. 64 No. 4 (2012), 1169-1180) they gave the first instance where Kähler or non-Kähler conditions of the image spaces make a difference in the value distribution theory. In this paper, we will investigate orders of meromorphic mappings into a Hopf surface which is more general than dealt with by Noguchi-Winkelmann, and an Inoue surface. They are non-Kähler surfaces and belong to VII₀-class. For a general Hopf surface S, we prove that there exists a differentiably non-degenerate holomorphic mapping f: C² → S with order at most one. For any Inoue surface S′, we prove that every non-constant meromorphic mapping f: Cⁿ → S′ is holomorphic and its order satisfies ρ_f ≥ 2.