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Power series with Hadamard gaps and hyperbolic complete minimal surfaces

DOI：10.13255/j.cnki.jusst.20210524002

 作者 单位 E-mail 张建肖 上海理工大学 理学院，上海 200093 刘晓俊 上海理工大学 理学院，上海 200093 xiaojunliu2007@hotmail.com

基于具有Hadamard缺项的特殊幂级数，研究了一类位于$\mathbb{R}^3$中2个平行平面之间的双曲型完备极小曲面族。首先得到如下结果：若$h(z) = \displaystyle\sum\limits_{j = 1}^\infty {{a_j}{z^{{n_j}}}}$是一个具有Hadamard缺项的幂级数，其中，$z \in {\mathbb{C}}$，$j = 1,2, \cdots$，且满足给定的3个特殊条件，则对于单位圆盘 $\Delta$内的任意发散曲线$\gamma$，有$\displaystyle\int_\gamma {{{\left| {h'(z)} \right|}^2}\left| {{\rm{d}}z} \right|} = \infty$。同时列举出了满足上述条件的具体的解析函数，其次通过选择适当的Weierstrass表示对，并利用上述结论，构造出了位于$\mathbb{R}^3$中2个平行平面之间的双曲型完备极小曲面族及其具体形式。

Based on the special power series with Hadamard gaps, a family of hyperbolic complete minimal surfaces located between two parallel planes in ${\mathbb{R}^3}$ was studied. Firstly, the following results were obtained: Let $h(z) = \displaystyle \sum\limits_{j = 1}^\infty {{a_j}{z^{{n_j}}}}$ be a series with Hadamard gaps, where $z \in \mathbb{C}$, $j = 1,2, \cdots$, and satisfy three given special conditions. Then for all divergent paths $\gamma$ in the unit disk $\Delta$, $\displaystyle \int_\gamma {{{\left| {h'(z)} \right|}^2}\left| {dz} \right|} = \infty$ satisfies. At the same time, the specific analytical functions satisfying the above conditions were listed. Secondly, by selecting the appropriate Weierstrass representation pair and using the above conclusion, the hyperbolic complete minimal curved surface family and its specific form between two parallel planes in ${\mathbb{R}^3}$ were constructed.
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