Theory of injective holomorphic mappings of a Riemann surface into another and its applications to hydrodynamics - a modern comprehension of the classical theory of univalent functions

黎曼曲面到另一个曲面的单射全纯映射理论及其在流体动力学中的应用 - 对单价函数经典理论的现代理解

• 批准号: 11440048
• 负责人: SHIBA Masakazu
• 金额: $ 7.87万
• 依托单位: HIROSHIMA UNIVERSITY
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 1999
• 资助国家: 日本
• 起止时间: 1999 至 2002
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-11440048/
• 关键词: Rigmann surfaces; Conformal embedding; Univalent functions; Analytic continuation; Fluid dynamics; Quadratic differentials; Fundamental domains; Hyperbolically maximal domains; 正則な単射; 自己等角写像群; 双曲的計量; 双曲的極大領域; ワイアシュトラスのゼータ関数; トーラスのモジュラス; 流れのエ

The principal aim of the project has been to generalize the classical theory of univalent functions to that of conformal embeddings of a Riemann surface into another, and to find a number of properties of such mappings which had never been observed in the classical case. We studied hydrodynamics on Riemann surfaces, particularly on a torus. We investigated a new type of extremal problem (with respect to the hyperbolic metric) to obtain the notion of "hyperbolically maximal domain". We proved that a hyperbolically maximal domain is a fundamental domain and its boundary consists of trajectory arcs of a meromorphic quadratic differential. We then gave a new construction of a fundamental domain for a Fuchsian group. We also studied the new concept of analytic continuation "beyond the ideal boundary".Part of the obtained results is reported in several international meetings: in Catania, Italy (2000), in Seoul, Korea (2001), Aveiro, Portugal (2001), and in Halle, Germany (2002).

该项目的主要目的是将单价函数的经典理论推广到黎曼曲面到另一个曲面的共形嵌入理论，并找到这种映射的许多在经典情况下从未观察到的属性。我们研究了黎曼曲面上的流体动力学，特别是圆环面上的流体动力学。我们研究了一种新型极值问题（相对于双曲度量）以获得“双曲最大域”的概念。我们证明了双曲极大域是一个基本域，其边界由亚纯二次微分的轨迹弧组成。然后我们给出了 Fuchsian 群的基本域的新构造。我们还研究了“超越理想边界”的解析延拓的新概念。所获得的部分结果在几个国际会议上报告：意大利卡塔尼亚（2000年）、韩国首尔（2001年）、葡萄牙阿威罗（2001年） ，以及德国哈勒（2002 年）。

项目成果

S.Kakei: "A Differential-Difference System Related to Toroidal Lie Algebra"J. Phys. A : Math. Gen.. 34・48. 10585-10592 (2001)

S.Kakei：“与环形李代数相关的微分系统”J. 10585-10592。

Mizuta, Y.: "Hyperplane integrals of BLD and monotone BLD functions"数理解析研究所講究録. 1116. 51-68 (1999)

Mizuta, Y.：“BLD 和单调 BLD 函数的超平面积分”数学研究所 Kokyuroku。1116. 51-68 (1999)

Mizuta, Y.: "Boundary limits of spherical means for BLD and monotone BLD functions in the unit ball"Acad. Sci. Fenn. Ser. A. I. Math.. 24. 45-60 (1999)

Mizuta, Y.：“单位球中 BLD 和单调 BLD 函数的球形装置的边界限制”Acad。

Nagamachi, S.: "Hyperfunction quantum field theory : Analytic structure, modular aspects and local observable algebras"Journal of Mathematical Physics. 42. 99-129 (2001)

Nagamachi, S.：“超函数量子场论：解析结构、模方面和局部可观测代数”数学物理杂志。

Chung, Y.M.: "Topological entropy for differentible maps of intervals"Osaka J. Math.. 38. 1-12 (2001)

Chung, Y.M.：“区间可微映射的拓扑熵”Osaka J. Math.. 38. 1-12 (2001)