Project/Area Number  09440063 
Research Category 
GrantinAid for Scientific Research (B)

Section  一般 
Research Field 
解析学

Research Institution  Yamaguchi University 
Principal Investigator 
MASUMOTO Makoto Faculty of Science, Yamaguchi University Associate Prof., 理学部, 助教授 (50173761)

CoInvestigator(Kenkyūbuntansha) 
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)
岡田 真理 山口大学, 工学部, 助教授 (40201389)
HATAYA Yasushi Faculty of Science, Yamaguchi University Research Associate, 理学部, 助手 (20294621)

Project Fiscal Year 
1997 – 1998

Project Status 
Completed(Fiscal Year 1998)

Budget Amount *help 
¥5,100,000 (Direct Cost : ¥5,100,000)
Fiscal Year 1998 : ¥2,200,000 (Direct Cost : ¥2,200,000)
Fiscal Year 1997 : ¥2,900,000 (Direct Cost : ¥2,900,000)

Keywords  Riemann surface / holomorphic mapping / conformal mapping / Teichmuller space / リーマン面 / 正則写像 / 等角写像 / タイヒシュラー空間 / タイヒミュラー空間 / 擬等角写像 
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 borderlike 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 borderlike boundary component, then condition (a) does not necessarily follow from condition (a).
