2016 Fiscal Year Final Research Report
New perspective on Riemann surfaces of infinite genus and their deformation spaces
| Project/Area Number |
26610014
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| Research Category |
Grant-in-Aid for Challenging Exploratory Research
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| Allocation Type | Multi-year Fund |
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| Research Field |
Geometry
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| Research Institution | Tokyo Institute of Technology |
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Principal Investigator |
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| Project Period (FY) |
2014-04-01 – 2017-03-31
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| Keywords | 関数論 / トポロジー / Hyperbolic geometry / Teichmuller space / Riemann surface |
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| Outline of Final Research Achievements |
On the deformation theory of Riemann surfaces, quasiconformal maps have played an important role. In this project, we succeeded in constructing a counterexample of Sullivan-Thurston’claim for quasiconformal motions. We also give a necessary and sufficient condition for a holomorphic motion of a subset of the Riemann sphere parametrized over a Riemann surface to be extended to a holomorphic motion of the Riemann sphere. The condition is given by the monodromy of the motion. This result is related to quasiconformal maps but it has also a topological aspect, the monodromy.
A new concept of Teichmuller space for Riemann surface with countably many points is considered. The research of those Teichmuller spaces will generalize holomorphic motions to those of subsets of Riemann surfaces. Therefore, it will give new interesting problems to related fields of mathematics.
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| Free Research Field |
数物系科学
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