Holomorphic curves with shift-invariant hyperplane preimages

TL;DR

This paper proves that certain holomorphic curves with shift-invariant hyperplane preimages are necessarily periodic, extending classical Picard theorems to a difference setting and introducing new tools like a difference Cartan's second main theorem.

Contribution

It introduces a difference analogue of Green's Picard theorem for holomorphic curves and develops new versions of Cartan's second main theorem and the lemma on logarithmic derivatives.

Findings

Holomorphic curves with shift-invariant hyperplane preimages are periodic.

New difference Cartan's second main theorem established.

Examples demonstrate the sharpness of the results.

Abstract

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 t(z)=z+c, then f is periodic with period 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.