Propagation sets of holomorphic curves

TL;DR

This paper introduces the concept of propagation sets for holomorphic curves, demonstrating how properties like uniqueness can extend from subsets to the entire complex plane, and establishes classical value theorems within these sets.

Contribution

The paper defines propagation sets for holomorphic curves and proves their existence for infinite and finite order cases, extending classical value theorems to these sets.

Findings

Existence of propagation sets as unions of disks or annuli.

Extension of five-value and four-value theorems to propagation sets.

Applicability to both finite and infinite order holomorphic curves.

Abstract

We consider a problem of whether a property of holomorphic curves on a subset $X$ of the complex plane can be extended to the whole complex plane. In this paper, the property we consider is uniqueness of holomorphic curves. We introduce the propagation set. Simply speaking, $X$ is a propagation set if linear relation of holomorphic curves on the part of preimage of hyperplanes contained in $X$ can be extended to the whole complex plane. If the holomorphic curves are of infinite order, we prove the existence of a propagation set which is the union of a sequence of disks (In fact, the method applies to the case of finite order). For a general case, the union of a sequence of annuli will be a propagation set. The classic five-value theorem and four-value theorem of R. Nevanlinna are established in such propagation sets.