1998 Fiscal Year Final Research Report Summary
Research on holomorphic mappings between Riemann surfaces
| Project/Area Number |
09440063
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
解析学
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| Research Institution | Yamaguchi University |
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Principal Investigator |
MASUMOTO Makoto Faculty of Science, Yamaguchi University Associate Prof., 理学部, 助教授 (50173761)
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| Co-Investigator(Kenkyū-buntansha) |
HATAYA Yasushi Faculty of Science, Yamaguchi University Research Associate, 理学部, 助手 (20294621)
KASHIWAGI Yoshimi Faculty of Economics, Yamaguchi University Associate Prof., 経済学部, 助教授 (00152637)
GOUMA Tomomi Faculty of Science, Yamaguchi University Research Associate, 理学部, 助手 (70253135)
YANAGIHARA Hiroshi Faculty of Engineering, Yamaguchi University Associate Prof., 工学部, 助教授 (30200538)
KATO Takao Faculty of Science, Yamaguchi University Prof., 理学部, 教授 (10016157)
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| Project Period (FY) |
1997 – 1998
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| Keywords | Riemann surface / holomorphic mapping / conformal mapping / Teichmuller space |
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| Research Abstract |
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).
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