If f : C → P n is a holomorphic curve of hyper-order less than one for which 2n + 1 hyperplanes in general position have forward invariant preimages with respect to the translation τ (z) = z + c, then f is periodic with period c ∈ C. This result, which can be described as a difference analogue of M. Green's Picard-type theorem for holomorphic curves, follows from a more general result presented in this paper. The proof relies on a new version of Cartan's second main theorem for the Casorati determinant and an extended version of the difference analogue of the lemma on the logarithmic derivatives, both of which are proved here. Finally, an application to the uniqueness theory of meromorphic functions is given, and the sharpness of the obtained results is demonstrated by examples.2000 Mathematics Subject Classification. Primary 32H30, Secondary 30D35.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.