The Algebras of Bounded Analytic Functions on a Riemann Surface and the isomorphic problem

黎曼曲面上所有有界解析函数的代数当然是由黎曼曲面决定的。这个匡威不再明显，我们称之为同构问题。如果一个黎曼曲面允许一个亚纯函数在曲面的边界处有界，并且有界解析函数分离曲面的点，那么答案是肯定的。然而，当曲面不存在这样的亚纯函数时，研究者发现了一个反例，当我们问同构问题时，我们假设有界解析函数（弱）分离曲面上的点，否则同构问题总是失败的。因此，它是一个有趣的问题，以确定是否一个给定的表面是分离或不关于有界解析函数。这个问题一般也很难解决。研究者与Mitsuru Nakai教授和Dc. Yasuyuki小林，考虑两片圆盘的情况，其中分支点序列的投影会聚到圆盘的中心。我们考虑了一列以分歧点为中心的小圆盘被去掉的情况，得到了分离和不分离的几个充分条件。

The algebra of all bounded analytic functions on a Riemann surface is of course determined by the Riemann surface. This converse is no longer obvious, and we call it the isomorphic problem. If a Riemann surface admits a meromorphic function bounded at the boundary of the surface and if bounded analytic functions separate the points of the surface, then the answer is yes. However, the investigator found a counter example when a surface admits no such meromorphic function.When we ask the isomorphic problem, we assume that bounded analytic functions (weakly) separate the points of a surface, for otherwise isomorphic problem fails always. Thus, it is an interesting problem to determine whether a given surface is separating or not with respect to bounded analytic functions. This problem is also hard to solve in general. The investigator, together with Prof. Mitsuru Nakai and Dc. Yasuyuki Kobayashi, consider the case of a two-sheeted disc, where the projection of the sequence of ramification points converging to the center of the disc. We considered the case that a sequence of small discs centered at ramification points are removed, and found several sufficient conditions for separating and also not separating.