2005 Fiscal Year Final Research Report Summary
Extremal problems and holomorphic mappings of Riemann surfaces
| Project/Area Number |
16540158
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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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 Yamaguchi University, Faculty of Science, Prof., 理学部, 教授 (50173761)
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| Co-Investigator(Kenkyū-buntansha) |
SHIBA Masakazu Hiroshima Univ., Graduate School of Engineering, Prof., 大学院・工学研究科, 教授 (70025469)
YAMADA Akira Tokyo Gakugei Univ., Faculty of Education, Prof., 教育学部, 教授 (60126331)
YANAGIHARA Hiroshi Yamaguchi University, Faculty of Engineering, Associate Prof., 工学部, 助教授 (30200538)
YANAGI Kenjiro Yamaguchi University, Faculty of Engineering Science, Prof., 工学部, 教授 (90108267)
HATAYA Yasushi Yamaguchi University, Faculty of Science, Assistant Prof., 理学部, 助手 (20294621)
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| Project Period (FY) |
2004 – 2005
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| Keywords | Riemann surface / properly discontinuous group / hyperbolically maximal domain / circularizable domain / quadratic differential / conformal mapping / quasi-order |
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| Research Abstract |
Let R be a Riemann surface and let Γ be a group of conformal automorphisms of R acting properly discontinuously on R. Hyperbolically maximal domains on R for Γ are obtained by solving some extremal problems. They are closely related to Γ-invariant meromorphic quadratic differentials on R. This fact leads us to a new class of domains on R, that of Γ-circularizable domains. These domains have many interesting properties. If D is a Γ-cirucularizable domain, then it is simply connected. The boundary ∂D is a branched polygon unless it reduces to a point. In fact, ∂D consists of horizontal trajectories and critical points of some Γ-invariant meromorphic quadratic differential on R. If D is not compact, then for some point p in D the domain D is precisely invariant under the stabilizer of p in the whole group Γ.
Now, every hyperbolically maximal domain for Γ is Γ-circularizable. Many interesting properties of hyperbolically maximal domains come from their circularizability. We give a necessary and sufficient condition for a Γ-circularizable domain to be hyperbolically maximal for Γ.
Next, we generalize the above mentioned extremal problems and obtain generalized hyperbolically maximal domains. These generalized domains turns out to be more naturally related to circularizable domains than the original ones.
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