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黎曼曲面与另一个黎曼曲面的共形嵌入以及粘性流体的流体力学。对于第一个主题，我们得到了黎曼曲面上的“可循环域”的概念，并证明了这些域在证明经典的黎曼映射定理中起着重要的作用，并为构造不连续的共形自同构群…的基本域提供了一个新的工具比如富克西亚族。第二个主题是基于这样的认识，即保角嵌入理论中的各种面积定理将与粘性流体在管内的流动密切相关。这一猜想在2006年秋天首次得到证实。我们已经成功地测定了管内流体的粘度，这实现了由首席研究员提出的绝对面积定理。由此我们得到了Poiseuille流的推广。结果已经在德国举行的国际会议上公布，秋季会议也在日本数学学会上公布。此外，作为共形嵌入理论的一个应用，我们确定了极值平行狭缝映射的复线性组合的一个充要条件，从而解决了Maitani问题。

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.