基于Hadamard缺项幂级数的完备极小曲面

A Study of Complete Very Small Surfaces Based on the Power Series of Hadamard’s Missing Term

摘要: 1992年，Brito利用特殊的Hadamard缺项幂级数构建了一族位于ℝ³中两平行平面间的完备极小曲面。但由于其定理的条件(2)要求过强，故2022年张建肖对此作出了相应的改进，但其定理仍有不足之处。本文主要针对其定理的条件(2)以及C_k的放缩方式作出进一步的改进，相比于张建肖定理的条件(2)，本文放缩更精确。本文借助Cauchy-Schwarz不等式对C_k进行放缩，通过选取特定的Weierstrass表示对，证明了对于任意的发散曲线γ，都有相同的结论成立，并举出了相应的例子。

Abstract: In 1992, Brito constructs a family of complete very small surfaces located between two parallel planes in the middle of ℝ³ by using a special power series of Hadamard’s missing term. However, due to the excessive requirement of condition (2) of the theorem, Zhang Jianxiao made corre-sponding improvements in 2022, but her theorem still has shortcomings. This paper focuses on the condition (2) of its theorem and the deflation of C_k to make further improvements, compared with the condition (2) of Zhang Jianxiao’s theorem, this paper is more precise than the condition (2) of Zhang Jianxiao’s theorem.. In this paper, with the help of the Cauchy-Schwarz inequality for the deflation of C_k, we prove that the same conclusion holds for any divergence curve γ by choosing a specific pair of Weierstrass representations, and we give the corresponding example.