2002 Fiscal Year Final Research Report Summary
Holomorphic mappings and moduli of Riemann surfaces
| Project/Area Number |
12440041
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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 |
Basic analysis
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| Research Institution | Yamaguchi University |
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Principal Investigator |
MASUMOTO Makoto Faculty of Science, Associate Prof., 理学部, 助教授 (50173761)
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| Co-Investigator(Kenkyū-buntansha) |
YAMADA Akira Tokyo Gakugei Univ., Faculty of Education, Prof., 教育学部, 教授 (60126331)
KATO Takao Faculty of Science, Prof., 理学部, 教授 (10016157)
SHIBA Masakazu Hiroshima Univ., Faculty of Engineering, Prof., 工学部, 教授 (70025469)
YANAGIHARA Hiroshi Faculty of Engineering, Associate Prof., 工学部, 助教授 (30200538)
SUZUKI Noriaki Nagoya Univ., Graduate School of Math., Associate Prof., 大学院・多元数理科学研究科, 助教授 (50154563)
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| Project Period (FY) |
2000 – 2002
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| Keywords | Riemann surface / conformal mapping / extremal length / properly discontinuous group / fundamental domain / quasi order |
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| Research Abstract |
Let R be a closed Riemann surface of positive genus g. In the previous research we have shown that the extremal lengths of homology classes satisfy an algebraic equation of degree 2g depending on the genus. In the present research we prove a more primitive algebraic equation of degree 2, which easily leads us to the previously shown equation. Moreover, applying the equation of degree 2, we find a new symmetry of closed Riemann surfaces of genus 2. Also, we give a criterion for an open Riemann surface of positive finite genus to belong the class O_<AD> in terms of the conjugate operator of the space of harmonic differentials.
Next, let Γ be a properly discontinuous group of conformal automorphisms of a Riemann surface R. We introduce a quasi order in a class of simply connected subdomains of R ; the quasi order is defined in terms of hyperbolic metrics on the simply connected subdomains. We show that the class has a unique maximal element. The maximum D is a locally finite fundamental domain for Γ. If R is simply connected, then any connected component of the boundary of D is piecewise analytic simple arc or curve. Furthermore, our fundamental domains are different from Dirichlet or Ford fundamental domains. We thus obtain a new natural method to construct a fundamental domain for Γ.
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