Holomorphic mappings and moduli of Riemann surfaces

黎曼曲面的全纯映射和模

• 批准号: 12440041
• 负责人: MASUMOTO Makoto
• 金额: $ 3.9万
• 依托单位: Yamaguchi University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 2000
• 资助国家: 日本
• 起止时间: 2000 至 2002
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-12440041/
• 关键词: Riemann surface; conformal mapping; extremal length; properly discontinuous group; fundamental domain; quasi order; 代数曲線; 擬等角写像; マルチン境界; ペアノ曲線; フェルマー曲線; 線形誤り訂正符号; ランキンの卵; アールフォース関数; 正則写像; タイヒミュラー空間; 双曲的長さ; 解析接続

Let R be a closed Riemann surface of positive genus g. In the previous research we have shown that the extremal lengths of homology classes satisfy an algebraic equation of degree 2g depending on the genus. In the present research we prove a more primitive algebraic equation of degree 2, which easily leads us to the previously shown equation. Moreover, applying the equation of degree 2, we find a new symmetry of closed Riemann surfaces of genus 2. Also, we give a criterion for an open Riemann surface of positive finite genus to belong the class O_<AD> in terms of the conjugate operator of the space of harmonic differentials.Next, let Γ be a properly discontinuous group of conformal automorphisms of a Riemann surface R. We introduce a quasi order in a class of simply connected subdomains of R ; the quasi order is defined in terms of hyperbolic metrics on the simply connected subdomains. We show that the class has a unique maximal element. The maximum D is a locally finite fundamental domain for Γ. If R is simply connected, then any connected component of the boundary of D is piecewise analytic simple arc or curve. Furthermore, our fundamental domains are different from Dirichlet or Ford fundamental domains. We thus obtain a new natural method to construct a fundamental domain for Γ.

设R是正亏格g的闭黎曼曲面。在以前的研究中，我们已经表明，同调类的极值长度满足一个代数方程的程度2g依赖于亏格。在本研究中，我们证明了一个更原始的代数方程的次数为2，这很容易导致我们前面所示的方程。此外，应用二次方程，我们找到了亏格为2的闭黎曼曲面的一个新的对称性。利用调和微分空间的共轭算子给出了有限亏格的开Riemann曲面属于O_类的一个判别准则<AD>.在R的一类单连通子域中引入了拟序，该拟序定义为单连通子域上的双曲度量。我们证明了这个类有唯一的极大元。最大值D是Γ的局部有限基本区域。如果R是单连通的，则D的边界的任何连通分支都是分段解析的简单弧或曲线。此外，我们的基本域不同于Dirichlet或福特基本域.从而得到了一种新的构造Γ的基本整环的自然方法。

项目成果

Noriaki Suzuki: "A characterization of heat balls by a mean value property for temperatures"Proceedings of the American Mathematical Society. 129巻9号. 2709-2713 (2001)

Noriaki Suzuki：“通过温度平均值来描述热球”，美国数学会论文集，第 129 卷，第 9 期，2709-2713 (2001)。

Tetsuhiko Miyoshi: "Direction and curvature of the cracks in two-dimensional elastic bodies"Japan Journal of Industrial and Applied Mathematics. 12卷2号. 295-307 (2000)

三好哲彦：“二维弹性体中裂纹的方向和曲率”日本工业与应用数学杂志第12卷第2期295-307（2000年）。

Makoto Masumoto: "Algebraic relations among extremal lengths of homology classes on compact Riemann surfaces"Mathematische Nachrichten. 239/240. 157-169 (2002)

Makoto Masumoto：“紧黎曼曲面上同调类的极值长度之间的代数关系”Mathematische Nachrichten。

Makoto Masumoto: "Hyperbolically maximal domains and fundamental domains for a discontinuous group of conformal automorphisms of a Riemann surface"The Quarterly Journal of Mathematics. 53. 337-346 (2002)

Makoto Masumoto：“黎曼曲面的不连续群共形自同构的双曲最大域和基本域”《数学季刊》。

Kiyoshi Kawazu: "A diffusion process with a one-sided Brownian potential"Tokyo Journal of Mathematics. 24巻1号. 211-229 (2001)

Kiyoshi Kawazu：“具有单侧布朗势的扩散过程”《东京数学杂志》，第 24 卷，第 1. 211-229 期（2001 年）。