Research on holomorphic mappings between Riemann surfaces

设R是正有限亏格的有标记开黎曼曲面。研究了R同伦到同胚的同亏格标闭黎曼曲面的空间H。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，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有标记的紧黎曼曲面的空间M进行比较。显然，M是H的子集。如果亏格大于1，且R与从紧黎曼曲面去掉离散集后得到的黎曼曲面共形等价，则M与H相同。另一方面，证明了如果R有边界似边界分支，则M是H的真子集。设R和S是彼此同胚的黎曼曲面。并将R的一个同胚映射h固定在H上。我们对下列条件感兴趣：(A)存在R到S同伦h的保形映射。(B)R到h的S同伦的保形映射。根据Schiffer定理，当K是模数有限的双连通平面黎曼曲面时，反之亦然。我们应用上一段中的结果来证明：如果R是正的有限亏格，并且有一个边界似边界分支，则条件(A)不一定来自条件(A)。

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).