Research on holomorphic mappings of Riemann surfaces----roles of handles played in the existence problem of holomorphic mappings

黎曼曲面是一维连通的复流形，在拓扑上是可定向的曲面，因此一般有几个手柄。从环面上去掉一点而得到的黎曼曲面同胚于环面的黎曼曲面称为一次孔环。曾经被打洞的环面形成了一个真正的三维空间。在本研究中，我们用带标记的句柄固定Riemann曲面Y。我们研究了一次孔环面X的集合A，其中存在X到Y的全纯映射，其像对应于所标记的句柄，并将A与一次孔环面X组成的子集进行比较，证明了A是B的闭包，B是A的真子集，除非Y是环面或一次孔环面.

A Riemann surface is a connected complex manifold of dimension one, which is topologically an orientable surface and hence has several handles in general. A Riemann surface homeomorphic to the surface obtained from a torus by removing one point is called a once-holed torus. The once-holed tori make a space of real dimension three. In the present research we fix a Riemann surface Y with a marked handle. We investigate the set A of once-holed tori X for which there is a holomorphic mapping X into Y whose image corresponds to the marked handle, and compare A with the subset consisting of once-holed tori X which allow holomorphic mappings of finite degree into Y. It turns out that A is the closure of B and that B is a proper subset of A unless Y is a torus or a once-holed torus.