2003 Fiscal Year Final Research Report Summary
Analysis of quasisymmetric functions and Teichmuller space
| Project/Area Number |
13640185
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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 | Osaka City University |
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Principal Investigator |
SAKAN Ken-ichi Osaka City University, Graduate School of Science, Associate Professor, 大学院・理学研究科, 助教授 (70110856)
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| Co-Investigator(Kenkyū-buntansha) |
小森 洋平 大阪市立大学, 大学院・理学研究科, 講師 (70264794)
NISHIO Masaharu Osaka City University, Graduate School of Science, Lecturer, 大学院・理学研究科, 助教授 (90228156)
IMAYOSHI Yoichi Osaka City University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (30091656)
NAKANISHI Toshihiro Nagoya University, Graduate School of Polymathematics, Associate Professor, 大学院・多元数理科学研究科, 助教授 (50701546)
TUGAWA Toshiyuki Hiroshima University, Graduate School of Science, Associate Professor, 大学院・理学研究科, 助教授 (30235858)
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| Project Period (FY) |
2001 – 2003
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| Keywords | quasisymmetric function / quasiconformal mapping / quasiconformal extension / extremal extension / crossratio / conjugte function / crossratio / conjugate function / harmonic extension / Teichmuller space |
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| Research Abstract |
The head investigator Sakan published jointly with the foreign joint worker D.Partyka "On pseudo-metrics on the space of generalized quasisymmetric automorphisms of a Jordan curve" and "On Heinz's inequality ". In the former paper they introduced, with no use of quasiconformal mappings, some pseudo-metrics into the space of generalized quasisymmetric automorphisms of a Jordan curve, and discussed some applications to topological properties of the Teichmiuller pseudo-metric. In the latter paper they generalized the Heinz's result on one-to-one harmonic mappings F of the unit disc onto itself in the case where F is the Poisson integral of a sense-preserving homeomorphic self-mapping of the unit circle. As an application they inferred a version of Heinz's inequality for harmonic and quasiconformal self-mappings of the unit disc. In a paper to be. submitted, they show an asymptotically sharp variant of Heinz's inequality for harmonic and quasiconformal self-mappings of the unit disc.
For the analysis of quasisymmetry, it is important to analyze the representations of maximal dilatations and so on in terms of harmonic measure and crossratio. Furthermore, it turned out that boundary dilatations, conjugate functions and Cauchy singular integrals are deeply related to the quasiconformality of harmonic extensions. Sakan has discussed about these analyses, with investigators Nishio and Yoshida from the viewpoint of potential and probability theory. Further, on extremal extensions which are quite related to boundary dilatations Sakan has contacted foreign joint workers Y.Shen, S.Wu and investigators Ohtake and Sugawa. Moreover, Sakan has discussed with investigators Imayoshi, Komori and Nakanishi about the relation of their researches on Teichmuller space and our research project.
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