全纯曲线正规族分担超平面

利用亚纯函数值分布理论和正规族理论、线性代数理论及研究方法，研究了全纯曲线族分担超平面的正规性。设\begin{document}$ \mathcal{F} $\end{document}是从\begin{document}$ D\subset \mathbb{C} $\end{document}到\begin{document}${\mathbb{P}}^{3}\left(\mathbb{C}\right) $\end{document}的一族全纯映射，\begin{document}$ {H}_{0}$\end{document}和\begin{document}${H}_{l}({H}_{l}\ne {H}_{0}) $\end{document}是\begin{document}$ {\mathbb{P}}^{3}\left(\mathbb{C}\right) $\end{document}上处于一般位置的超平面，\begin{document}$l=1,2,\cdots,8 $\end{document}。假定对于任意的\begin{document}$ f\in \mathcal{F} $\end{document}满足条件：\begin{document}$f(\textit{z})\in H_l$\end{document}当且仅当\begin{document}$\nabla f \in H_l=\{x\in {\mathbb{P}}^{3}\left(\mathbb{C}\right): $\end{document}\begin{document}$ \langle x, \alpha_l \rangle=0\}$\end{document}；若\begin{document}$f(\textit{z})\in H_l $\end{document}的并集，有\begin{document}$|\langle f\left(z\right),{H}_{0}\rangle|/(\|f\|\|{H}_{0}\|)$\end{document}大于或等于\begin{document}$\delta $\end{document}。\begin{document}$0 < \delta < 1 $\end{document}，\begin{document}$\delta $\end{document}是常数，则 \begin{document}$ \mathcal{F} $\end{document}在D上正规。

Shared hyperplanes and normal families of holomorphic curves

Based on some fundamental knowledge, research methods and results about the theories of value distribution and normal family for meromorphic functions, and linear algebra, the normality of the families of holomorphic curves is considered. Let \begin{document}$ \mathcal{F} $\end{document} be a family of holomorphic maps of a domain \begin{document}$ D\subset \mathbb{C} $\end{document} into \begin{document}$ {\mathbb{P}}^{3}\left(\mathbb{C}\right). $\end{document} Let \begin{document}$ {H}_{0} $\end{document} and \begin{document}$ {H}_{l}\ne {H}_{0} $\end{document} be hyperplanes in \begin{document}$ {\mathbb{P}}^{3}\left(\mathbb{C}\right) $\end{document} located in general position, where \begin{document}$ l=1,2,\cdots,8 $\end{document}. Assume the following conditions hold for every \begin{document}$ f\in \mathcal{F}:f\left(z\right) $\end{document} belongs to \begin{document}$ {H}_{l} $\end{document}; if and only if \begin{document}$ \nabla f $\end{document} belongs to \begin{document}${H}_{l} =\{x\in {\mathbb{P}}^{3}\left(\mathbb{C}\right): \langle x, \alpha_l \rangle=0\}$\end{document}; if \begin{document}$ f\left(z\right) $\end{document} belongs to the union set of \begin{document}${H}_{l}$\end{document}, then \begin{document}$|\langle f\left(z\right),{H}_{0}\rangle|/ $\end{document}\begin{document}$ \|f\|\|{H}_{0}\|$\end{document} is equal or greater than \begin{document}$ \delta $\end{document}, where \begin{document}$ 0 < \delta < 1 $\end{document} is a constant. Then \begin{document}$ \mathcal{F} $\end{document} is normal on.