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。这种反面不再明显，我们称之为同构问题。如果黎曼曲面允许有一个亚纯函数有界于该曲面的边界，并且如果有界解析函数将该曲面的点分开，则答案是肯定的。然而，当曲面不允许这样的亚纯函数时，研究者发现了一个反例。当我们提出同构问题时，我们假设有界解析函数(弱)分离曲面的点，否则同构问题总是失败的。因此，确定给定曲面关于有界解析函数是否分离是一个有趣的问题。这个问题也很难从总体上解决。研究人员与中井三郎教授和DC。Yasuyuki Kobayashi，考虑两片圆盘的情形，其中分支点序列的投影收敛到圆盘的中心。我们考虑了以分支点为中心的小圆盘序列被去掉的情况，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.