1999 Fiscal Year Final Research Report Summary
Research on projective structures Riemaun surfaces from analytic and geometric vieupo
| Project/Area Number |
10640160
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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 | KYUSHU UNIVERSITY (1999)
Nagoya University (1998) |
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Principal Investigator |
TANIGAWA Harumi Graduate School of Mathematics kyushu university associate professor, 大学院・数理学研究科, 助教授 (30236690)
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| Co-Investigator(Kenkyū-buntansha) |
谷川 晴美 九州大学, 大学院・数理学研究科, 助教授 (30236690)
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| Project Period (FY) |
1998 – 1999
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| Keywords | Riemany surfaces / projective structures / Kleinian groups / measured laminations / hyperbolic spaces |
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| Research Abstract |
In has been known that the space of projective structures on Riemann surfaces has a structure of a vector bundle. This classical fact is based on the fact that a projective structure on a Riemann surface is closely related to a certain differential equation. Therefore, most researches on projective structures have been from analytic viewpoint. However, in mid 80's, Thurston showed that "any projective structure is obtained by grafting some measured lamination to some hyperbolic structure".
The aim of this research has been to investigate projective structures from the above new geometric viewpoint. The classical analytic approach naturally keeps eyes on the underlying complex structure, namely, restrict the observation to each fiber over a complex structure. However, in the course of our research, it turned out that the global observation with the Thurston's geometric parametrization is often more useful than restricting the observation to each fiber, and furthermore, the global observation in turns gives some information about each fiber.
Our approach successfully find out new results, which are interacting each other, can be summed up to the following three points: 1) we showed that on any complex structure, there exist infinitely many exotic structures with fuchsian monodromies and a certain characterization of such structures, 2) in relation to (1), we gave some observation how projective structures with discrete monodromies distribute on each fibers, 3) we proved that for each fiber, the mapping sending each projective structure to its monodromv is proper whenever restricted to anv fiber.
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