2003 Fiscal Year Final Research Report Summary
Dynamics of modular groups on infinite dimensional Teichmuller spaces
| Project/Area Number |
14540156
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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 | Ochanomizu University |
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Principal Investigator |
MATSUZAKI Katsuhiko Ochanomizu Univ., Faculty of Science, Associate Prof., 理学部, 助教授 (80222298)
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| Co-Investigator(Kenkyū-buntansha) |
SUGAWA Toshiyuki Hiroshima Univ., Grad.School of Science, Associate Prof., 大学院・理学研究科, 助教授 (30235858)
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| Project Period (FY) |
2002 – 2003
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| Keywords | Teichmuller space / modular group / Riemann surface / quasiconformal map / hyperbolic geometry / Schwarzian derivative / Bers embedding / univalent function |
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| Research Abstract |
Teichmueller spaces are not homogeneous spaces and their mudular groups do not act transitively. For compact Riemann surfaces, modular groups act discontinuously, but this is not the case for infinite dimensional Teichmueller spaces. We study the moduli spaces of Riemann surafces of infinite type by considering the chaotic behavior of the action of modular groups. For a viewpoint of general topology, the moduli space is either metrizable or not of the first separation axiom. However, except for a singular part, it can possess a certain geometric structure. In this research, we characterize this stable region by hyperbolic geometric structure of a Riemann surface and construct a contracted moduli space by the completion of the stable region. Consequently, we can describe the closure of a point set in terms of the geomery of Riemann surfaces, which is a point of teh contracted module space.
We considered the space of pre-Schwarzian derivatives of univalent functions on the unit disk which extends to quasiconformal mappings of the extended plane in order to investigate the relation between connected components of the pre-Schwarzian derivatives of univalent functions on the unit disk which extends to quasiconformal mappings of the extended plane in order to investigate the relation between connected components of the pre-Schwarzian model of the universal Teichmueller space and classical families of univalent functions. We also investigated geometric properties of univalent functions with a prescribed growth of the Schwarzian derivative and found that they are starlike or convex according to the distance to the origin in the Bers embedding of the universal Teichmueller space.
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