2022 Fiscal Year Final Research Report
A new perspective of complex manifolds from the view point of generalizations of holomorphic motions
| Project/Area Number |
18K18717
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| Research Category |
Grant-in-Aid for Challenging Research (Exploratory)
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| Allocation Type | Multi-year Fund |
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| Review Section |
Medium-sized Section 12:Analysis, applied mathematics, and related fields
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| Research Institution | Kyoto Sangyo University (2019-2022)
Tokyo Institute of Technology (2018) |
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Principal Investigator |
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| Project Period (FY) |
2018-06-29 – 2023-03-31
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| Keywords | Holomorphic motion / Quasiconformal mapping / Riemann surface |
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| Outline of Final Research Achievements |
Holomorphic motions are defined as holomorphic families of injections on subset of the Riemann sphere parametrized by complex manifolds. On this topic, we give a counter example to Chirka's statement, and also an example of a holomorphic motion which satisfies a kind of a topological trivial condition but can not be extended to a holomorphic motion over the Riemann sphere.
We find some conditions for generalized Cantor sets to be quasiconformally equivalent to each other. We also give estimate maximal dilatations. A necessary and sufficient condition for a generalized Cantor set to be quasiconformally equivalent to the standard Cantor set is obtained. Those results imply that different Cantor sets are connected via holomorphic motions.
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| Free Research Field |
Complex Analysis
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| Academic Significance and Societal Importance of the Research Achievements |
The holomorphic motion is a quite simple object in mathematics, that is, it is a holomorphic family of injections on a set in the complex plane. We have found various aspects on holomorphic motions and quasiconformal mappings.
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