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Research on holomorphic mappings between Riemann surfaces

黎曼曲面间的全纯映射研究

基本信息

批准号：
09440063
负责人：
MASUMOTO Makoto
金额：
$ 3.26万
依托单位：
Yamaguchi University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
1997
资助国家：
日本
起止时间：
1997 至 1998
项目状态：
已结题

项目摘要

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).

设R为正有限属的标记开黎曼曲面。我们研究了具有同属标记的闭黎曼曲面的空间H，其中存在一个R同伦到一个同胚的全纯映射。空间H是Teichmuller空间T的一个子集。我们首先证明了当格为1时H与T重合，而当格大于1时H是T的紧子集。接下来，我们比较H与空间M的标记紧致黎曼曲面的同属，其中R可以共形嵌入。显然，M是h的一个子集。如果属大于1，并且R共形等价于一个由紧黎曼曲面去掉离散集得到的黎曼曲面，则M与h相同。另一方面，我们证明，如果R有一个类边边界分量，则M是h的一个固有子集。现在，设R和S是彼此同胚的黎曼曲面。将R的同胚h固定到S上，我们感兴趣的是以下条件：(a) R有一个正形映射到S同伦h (b) R有一个正形映射到S同伦h，条件(a)暗示条件(b)是平凡的。根据Schiffer定理，当K是有限模的双连通平面黎曼曲面时，反之也成立。我们应用前一段的结果来证明，如果R是正有限属并且有一个类边边界分量，则条件(a)不一定从条件(a)推导出来。