Study on the theory of conformal embeddings of a Riemann surface focused on the hyperrbolic metric and hydrodynamics of viscous fluids

我们的主要目的是研究共形嵌入的黎曼曲面到另一个和流体力学的粘性流体。关于第一个主题，我们得到的概念“circularizable域”的黎曼曲面，并表明，这些领域发挥了重要作用，证明经典的黎曼映射定理，并给予一个新的工具，在建设基本领域的不连续组的共形自同构.如Fuchsian群。第二个主题是基于共形嵌入理论中的各种面积定理将与管内粘性流体流动密切相关的认识。这个猜想在2006年秋天首次得到证实。我们已经成功地确定了流体的粘度在一个管实现了绝对面积定理由于首席调查员。我们由此得到了Poiquilille流的推广。研究结果已在德国举行的国际会议上公布，并在秋季会议上，日本数学学会。此外，作为共形嵌入理论的一个应用，我们确定了极值平行缝映射复线性组合的一个充要条件，从而给出了一个Maitani问题的解.

Our main purpose was to study conformal embeddings of a Riemann surface into another and hydrodynamics of viscous fluids. Concerning the first topic, we obtained the notion of "circularizable domains" on a Riemann surface and showed that these domains play an important role in the proof of classical Riemann mapping theorem and give a new tool in the construction of fundamental domains for a discontinuous group of conformal automorphisms … such as Fuchsian groups. The second topic is based on the insight that the various area theorems in the theory of conformal embedding will be closely connected with the viscous fluid flows in a tube. The conjecture was first verified in the fall of 2006. We have succeeded in determining the viscosity of the fluid in a tube which realizes the absolute area theorem due to the head investigator. We have thus obtained generalization of the Poiseuille flows. The results have been announced in an international meeting held in Germany and also in the Fall Meeting, Mathematical Society of Japan. Besides, as an application of the theory of conformal embedding, we determined a necessary and sufficient condition for a complex linear combination of extremal parallel slit mappings, which gives a solution to a problem of Maitani.