We consider a problem of whether a property of holomorphic curves on a subset [Formula: see text] of the complex plane can be extended to the whole complex plane. In this paper, the property we consider is the uniqueness of holomorphic curves. We introduce the propagation set. Simply speaking, [Formula: see text] is a propagation set if linear relation of holomorphic curves on the part of preimage of hyperplanes contained in [Formula: see text] 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 Nevanlinna are established in such propagation sets.

中文翻译：

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全纯曲线的传播集

我们考虑一个问题：复平面的一个子集[公式：见正文]上的全纯曲线的性质是否可以推广到整个复平面。在本文中，我们考虑的性质是全纯曲线的唯一性。我们介绍传播集。简单地说，[公式：见文]是一个传播集，如果[公式：见文]中包含的超平面原像部分的全纯曲线的线性关系可以推广到整个复平面。如果全纯曲线是无限阶的，我们证明了传播集的存在，它是一个磁盘序列的并集。（实际上，该方法适用于有限阶的情况。）对于一般情况，环序列的并集将是一个传播集。