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

该项目的主要目的是将经典的一元函数理论推广到黎曼曲面的共形嵌入理论，并找到一些在经典情况下从未观察到的这种映射的性质。我们研究了黎曼曲面上的流体力学，特别是环面上的流体力学。研究了一类新的关于双曲度规的极值问题，得到了“双曲极大域”的概念。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构成。然后给出了Fuchsian群的基本定义域的新构造。我们还研究了解析延拓的新概念“超越理想边界”。部分获得的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）和德国哈雷（2002年）。

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).