Research on holomorphic mappings between Riemann surfaces

设R是正有限亏格的标记开黎曼曲面。本文研究了由同一亏格的标记闭Riemann曲面构成的空间H，其中存在R同伦到同胚的全纯映射。空间H是Teichmuller空间T的一个子集。我们首先证明了如果亏格为1，则H与T重合，而如果亏格大于1，则H是T的紧子集。接下来，我们将H与R可以共形嵌入的同一亏格的标记紧致Riemann曲面空间M进行比较。显然，M是H的一个子集。如果亏格大于1，R共形等价于一个从紧致Riemann曲面通过去除离散集得到的Riemann曲面，则M与H相同。另一方面，我们证明了，如果R具有边界样边界分支，则M是H的一个真子集。现在，设R和S是彼此同胚的Riemann曲面，并将R的一个同胚h固定到S上。我们感兴趣的是下列条件：（a）存在R到S的一个同伦到h的共形映射。(b)存在R到S的一个与h同伦的共形映射。条件（a）蕴涵条件（B）是平凡的。根据Schiffer的一个定理，当K是一个具有有限模的双连通平面Riemann曲面时，其匡威也成立。我们应用前一段中的结果来证明，如果R是正有限亏格的，并且有一个border-like边界分支，则条件（a）不一定由条件（a）得出。

Let R be a marked open Riemann surface of positive finite genus. We are concerned with the space H of marked closed Riemann surfaces of the same genus into which there is a holomorphic mapping of R homotopic to a homeomorphism. The space H is a subset of the Teichmuller space T.We first show that H coincides with T if the genus is one, while H is a compact subset of T if the genus is greater than one. Next we compare H with the space M of marked compact Riemann surfaces of the same genus into which R can be conformally embedded. Obviously, M is a subset of H.If the genus is greater than one and R is conformally equivalent to a Riemann surface obtained from a compact Riemann surface by removing a discrete set, then M is identical with H.We prove, on the other hand, that if R has a border-like boundary component, then M is a proper subset of H.Now, let R and S be Riemann surfaces homeomorphic to each other, and fix a homeomorphism h of R onto S.We are interested in the following conditions :(a) There is a conformal mapping of R into S homotopic to h.(b) There is a conformal mapping of R into S homotopic to h.It is trivial that condition (a) implies condition (b). By a theorem of Schiffer, in the case where K is a doubly connected planar Riemann surface with finite modulus, the converse is also true. We apply the results in the preceding paragraph to show that if R is of positive finite genus and has a border-like boundary component, then condition (a) does not necessarily follow from condition (a).