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无数函数的经典理论推广到riemann表面的共形嵌入到另一个函数中，并找到在经典情况下从未观察到的这种映射的许多属性。我们研究了黎曼表面的流体动力学，尤其是在圆环上。我们研究了一种新型的极端问题（相对于双曲线指标），以获得“夸张最大结构域”的概念。我们证明了一个夸张的最大结构域是一个基本结构域，其边界由Meromormormormormormormormormormormormormortic二差分的轨迹组成。然后，我们为紫红色团体提供了一个新的建筑。我们还研究了分析延续的新概念“超出理想边界”。在几次国际会议上报告了所获得的结果的部分：在意大利卡塔尼亚（2000年），韩国首尔（2001），葡萄牙Aveiro（2001）和德国哈雷（2002）。

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。

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

Y-M.Chung：“区间微分映射的拓扑熵”Osaka J. Math.. 38. 1-12 (2001)

S.Nagamachi: "Hyperfunction quantum field theory : Analytic structure, modular aspects, and local observable algebras"J. Math. Phys.. 42・1. 99-129 (2001)

S.Nagamachi：“超函数量子场论：解析结构、模方面和局部可观测代数”J. Math. 99-129。

T.Futamura: "A generalization of Boche's theorem for polyharmonic functions"Hiroshima Math. J.. 31. 59-70 (2001)

T.Futamura：“Boche 多调和函数定理的推广”广岛数学。

Ramani, A.: "Linearizable mappings and the low-growth criterion"J. Phys. A: Math. Gen.. 33. L287-L292 (2000)

Ramani, A.：“线性化映射和低增长标准”J.