上海理工大学学报  2022, Vol. 44 Issue (4): 364-367 PDF

Power series with Hadamard gaps and hyperbolic complete minimal surfaces
ZHANG Jianxiao, LIU Xiaojun
College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China
Abstract: 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.
Key words: power series with Hadamard gaps     divergence curve     completeminimal surface     Weierstrass representation pair
1 问题的提出

1980年，Jorge等[2]利用Runge逼近定理证明了存在位于 ${\mathbb{R}^3}$ 中2个平行平面之间非平坦的完备极小曲面，从而否定了猜想1。

1996年，Nadirashvili[3]利用Runge逼近定理否定了猜想2，证明了存在极小浸入到 ${\mathbb{R}^3}$ 中单位球的具有负Gauss曲率的完备极小曲面。但是，他们的证明只是表明存在相应的极小曲面，没有给出具体的满足条件的极小曲面的例子。

1992年，Brito[4]利用Hadamard缺项幂级数构造了 ${\mathbb{R}^3}$ 中位于2个平行平面间的完备极小曲面族，给出了实例，得到定理1。

a. $\displaystyle\sum\limits_{j = 1}^\infty {\left| {{a_j}} \right|}$ 收敛；

b. $\mathop {\lim }\limits_{j \to \infty } \left| {{a_j}} \right|\min \left\{ {\left( {{{{n_j}} \mathord{\left/ {\vphantom {{{n_j}} {{n_{j - 1}}}}} \right. } {{n_{j - 1}}}}} \right),\left( {{{{n_{j + 1}}} \mathord{\left/ {\vphantom {{{n_{j + 1}}} {{n_j}}}} \right. } {{n_j}}}} \right)} \right\} = \infty$

c. $\displaystyle\sum\limits_{j = 1}^\infty {{{\left| {{a_j}} \right|}^2}{n_j}}$ 发散；

 $\int_\gamma {{{\left| {h'(z)} \right|}^2}\left| {{\rm{d}}z} \right|} = \infty$

a. $\left|\dfrac{{a}_{j+1}}{{a}_{j}}\right|⩽\eta \lt 1,\left|{a}_{j}\right|\ne 0$

b. 对于充分大的 $k \in {{\mathbb{Z}}^ + }$

 $\begin{split} &\displaystyle\sum\limits_{j = 1}^{k - 1} {\left| {{a_j}} \right|{n_j}} \lt \dfrac{1}{{4{\rm{e}}}}\left| {{a_k}} \right|{n_k}且 \dfrac{{q}_{k}}{2}-\mathrm{ln}{q}_{k}⩾1-\mathrm{ln}\dfrac{1-\eta }{8\eta } \end{split}$

c. $\displaystyle\sum\limits_{j = 1}^\infty {{{\left| {{a_j}} \right|}^2}{n_j}}$ 发散；

 $\int_\gamma {{{\left| {h'(z)} \right|}^2}\left| {{\rm{d}}z} \right|} = \infty$
2 定义与符号

$\Delta$ 为复平面 ${\mathbb{C}}$ 中的单位圆盘，现讨论由 $\Delta$ 参数化的完备极小曲面。

 ${n}_{j+1}/{n}_{j}\geqslant q \gt 1,\;\;j=1,2,\cdots \text{，}$

 $\left\{\begin{split} &{x_1}(z) = {a_1} + {{\rm{Re}}} \displaystyle\int_0^z {\dfrac{1}{2}f(1 - {g^2}){\rm{d}}\xi } \\ &{x_2}(z) = {a_2} + {{\rm{Re}}} \displaystyle\int_0^z {\dfrac{i}{2}f(1 + {g^2}){\rm{d}}\xi } \\ &{x_3}(z) = {a_3} + {{\rm{Re}}} \displaystyle\int_0^z {fg} {\rm{d}}\xi \end{split}\right.$ (1)

 ${\rm{d}}s = \lambda (z)\left| {{\rm{d}}z} \right| = \frac{1}{2}\left| f \right|(1 + {\left| g \right|^2})\left| {{\rm{d}}z} \right|$ (2)
3 定理2的证明

 ${R}_{k}=\left\{z\in \Delta :1-\frac{1}{{n}_{k}}\leqslant \left|z\right|\leqslant 1-\frac{1}{2{n}_{k}}\right\}$

 $\begin{split} & \left| {h'(z)} \right| = \left| {\displaystyle\sum\limits_{j = 1}^\infty {{n_j}{a_j}{z^{{n_j} - 1}}} } \right| \gt \left| {\displaystyle\sum\limits_{j = 1}^\infty {{n_j}{a_j}{z^{{n_j}}}} } \right| \geqslant \\ &\;\;{n}_{k}\left|{a}_{k}\right|{\left|z\right|}^{{n}_{k}}-\left|{\displaystyle \sum _{j=1}^{k-1}{n}_{j}{a}_{j}{z}^{{n}_{j}}}\right|-\left|{\displaystyle \sum _{j=k+1}^{\infty }{n}_{j}{a}_{j}{z}^{{n}_{j}}}\right|,\;z\in \Delta \end{split}$ (3)

 $\left|{h}^{\prime }(z)\right|\geqslant \left|{A}_{k}\right|-\left|{B}_{k}\right|-\left|{C}_{k}\right| \text{，}z \in \Delta \text{，}k \in {\mathbb{N}}$ (4)

 $\left|{A}_{k}\right|\geqslant \frac{1}{2{\rm{e}}}\left|{a}_{k}\right|{n}_{k} \text{，} k \geqslant {k}_{1} \text{，}z \in {R_k}$ (5)

 $\left| {{B_k}} \right| \lt \frac{1}{{4{\rm{e}}}}\left| {{a_k}} \right|{n_k} = \frac{1}{2}\left(\frac{1}{{2{\rm{e}}}}\left| {{a_k}} \right|{n_k}\right)\text{，} k\geqslant {k}_{2} \text{，}z \in {R_k}$ (6)

 $\begin{split} &\left|{C}_{k}\right|\leqslant {\displaystyle \sum _{j=k+1}^{\infty }\left|{a}_{j}\right|{n}_{j}{\left(1-\frac{1}{2{n}_{k}}\right)}^{{n}_{j}}} = \sum\limits_{j = k + 1}^\infty {\left| {{a_j}} \right|{n_j}{{\rm{e}}^{{n_j}\ln (1 - \frac{1}{{2{n_k}}})}}} \leqslant\\ &\quad\quad {\displaystyle \sum _{j=k+1}^{\infty }{\eta }^{j-k}\left|{a}_{k}\right|{n}_{j}{{\rm{e}}}^{-\frac{{n}_{j}}{2{n}_{k}}}} \text{，}z \in {R_k} \end{split}$

 $\left|{C}_{k}\right|\leqslant {\displaystyle \sum _{j=k+1}^{\infty }{\eta }^{j-k}\left|{a}_{k}\right|{n}_{k+1}{{\rm{e}}}^{-\frac{{n}_{k+1}}{2{n}_{k}}}},\;\;z\in {R}_{k}$ (7)

 $\begin{split} \left|{C}_{k}\right|\leqslant & \frac{\eta }{1-\eta }\left|{a}_{k}\right|{n}_{k+1}{{\rm{e}}}^{-\frac{{n}_{k+1}}{2{n}_{k}}} \leqslant \frac{1}{8{\rm{e}}}\left|{a}_{k}\right|{n}_{k} = \frac{1}{4}\left(\frac{1}{{2{\rm{e}}}}\left| {{a_k}} \right|{n_k}\right)\text{，} \\ &k\geqslant {k}_{1} \text{，}z \in {R_k}\\[-12pt] \end{split}$ (8)

 $\left|{h}^{\prime }(z)\right|\geqslant \frac{1}{8{\rm{e}}}\left|{a}_{k}\right|{n}_{k}$

$\Delta$ 内的发散曲线 $\gamma$ ，对于任意的 $k\geqslant l$ $l \in{\mathbb{N}}$ $\gamma$ 必定穿过 ${R_k}$ ，则

 $\begin{split} {{\displaystyle {\int }_{\gamma }\left|{h}^{\prime }(z)\right|}}^{2}\left|{\rm{d}}z\right|\geqslant & {\displaystyle \sum _{k=l}^{\infty }{\displaystyle {\int }_{\gamma \cap {R}_{k}}{\left|{h}^{\prime }(z)\right|}^{2}\left|{\rm{d}}z\right|}} \geqslant\\ &{\displaystyle \sum _{k=l}^{\infty }\frac{1}{128{{\rm{e}}}^{2}}{\left|{a}_{k}\right|}^{2}{n}_{k}} = \infty \end{split}$

$h(z) = \displaystyle\sum\limits_{j = 1}^\infty {{a_j}{z^{{n_j}}}}$ $z \in {\mathbb{C}}$ ，其中， ${a_j} = \dfrac{1}{{{2^j}}}, \; {n_j} = {(16{\rm{e}})^j}$ 。易得 $h(z)$ 满足定理2的条件a～c，但不满足定理1的条件b。

4 推　论

$A(\Delta )$ 是由在单位圆盘 $\Delta$ 内的解析的函数构成的集合。

 $\begin{split} {x}_{3}(z)=& \mathrm{Re}{\displaystyle {\int }_{0}^{z}f(\xi )g(\xi ){\rm{d}}\xi }=\\ &\mathrm{Re}{\displaystyle {\int }_{0}^{z}{h}^{\prime }(\xi ){\rm{d}}\xi }=\\ &\mathrm{Re}(h(z)-h(0))=\\ &\mathrm{Re}(h(z))\leqslant {\displaystyle \sum _{j=1}^{\infty }{a}_{j}{z}^{{n}_{j}}}\leqslant {\displaystyle \sum _{j=1}^{\infty }\left|{a}_{j}\right|} \lt \infty \end{split}$

 [1] CALABI E. The space of Kahler metrics[C]// Proceedings of the International Congress of Mathematics. Amsterdam, 1954 (2): 206-207. [2] JORGE L, XAVIER F. A complete minimal surface in ${\mathbb{R}^3}$ between two parallel planes [J]. Annals of Mathematics, 1980, 112(2): 203-206. [3] NADIRASHVILI N. An application of potential analysis to minimal surfaces[J]. Moscow Mathematical Journal, 2001, 1(4): 601-604. DOI:10.17323/1609-4514-2001-1-4-601-604 [4] DE BRITO F F. Power series with Hadamard gaps and hyperbolic complete minimal surfaces[J]. Duke Mathematical Journal, 1992, 68(2): 297-300. [5] 孙道椿. 缺项及随机级数的边界性质[J]. 武汉大学学报(自然科学版), 1991(1): 7-10. [6] 杨连中, 王安斌. 具有Hadamard缺陷的幂级数与正规级数(英文版)[J]. 东北数学, 1991, 7(4): 453-456. [7] GNUSCHKE D. On power series with Hadamard gaps[J]. Analysis, 1984, 4(1/2): 61-72. [8] ZYGMUND A. Trigonometric series[M]. Cambridge: Cambridge University Press, 1968. [9] 刘计善. 一类极小曲面造型设计的新方法[J]. 复旦学报(自然科学版), 2007, 46(2): 192-197. [10] XAVIER F, 潮小李. 现代极小曲面讲义[M]. 北京: 高等教育出版社, 2011. [11] OSSERMAN R. A survey of minimal surfaces[M]. New York: Dover Publications, 1986. [12] COSTA C J, SIMÖES P A Q. Complete minimal surfaces of arbitrary genus in a slab of ${\mathbb{R}^3}$ [J]. Annales de L'institut Fourier, 1996, 46(2): 535-546. DOI:10.5802/aif.1523 [13] BARBOSA J L M, COLARES A G. Minimal surfaces in $\mathbb{R}^3$ [J]. Lecture Notes in Mathematics, 1986, 27(1): 34-46.