1999 Fiscal Year Final Research Report Summary
Various quasiconformal extensions of a quasisymmetric automorphism of the unit circle and Teichmuller space
| Project/Area Number |
10640181
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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 Kenichi Osaka City University, Faculty of Science, Associate Professor, 理学部, 助教授 (70110856)
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| Co-Investigator(Kenkyū-buntansha) |
KOMORI Yohei Osaka City University, Faculty of Science, Assistant, 理学部, 助手 (70264794)
NISHIO Masaharu Osaka City University, Faculty of Science, Associate Professor, 理学部, 助教授 (90228156)
IMAYOSHI Masaharu Osaka City University, Faculty of Science, Professor, 理学部, 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)
SUGAWA Toshiyuki Kyoto University, Graduate School of Science, Assistant, 大学院・理学研究科, 助手 (30235858)
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| Project Period (FY) |
1998 – 1999
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| Keywords | quasisymmetric automorphism / quasiconformal extension / harmonic extension / extremal extension / Beurling-Ahlfors extension / Douady-Earle extension / generalized quasisymmetric dilatation / Teichmuller space |
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| Research Abstract |
1. The head investigator Sakan obtained jointly with Partyka and Zajac the following results in three published paper.
(1) They gave necessary and sufficient conditions on sense-preserving homeomorphisms of the unit circle for the quasiconformality of their harmonic extensions. In particular, the case where the harmonic extension is not quasiconformal was studied. In consequence, suitable examples were constructed.
(2) Suppose that the harmonic extension of a sense-preserving homeomorphism of the unit circle is quasiconformal. All such homeomorphisms with a bounded derivative were well characterized. In consequence, a generalization of Martio's result was obtained.
(3) By the use of the extension operators, a unified summary on harmonic and quasiconformal extensions was given.
2. Sakan and an investigator Sugawa attended the 12th Conference on Analytic Functions (August30〜September 5, 1998 at Lublin, Poland) and gave lectures. The papers of the results were published. Sakan expressed the Beurling-Ahlfors condition on quasisymmetry in terms of harmonic measure and cross ration. In consequence, Sakan introduced a generalized conformally invariant dilatation on quasisymmetry and discussed its properties. By an application of holomorphic families of univalent functions, Sugawa obtained some results on quasiconformal extendability of univalent functions.
3. Sakan attended the Second ISAAC Conference (August 16〜21, 1999 at Fukuoka Institute of Technology) and Korea-Japan Seminar on Complex Analysis (October 18〜20, 1999 at Yeungnam University, Korea) and gave lectures. The result will soon appear in the Proceedings. On the space of all quasisymmetric automorphisms of a given Jordan curve, a conformally invariant pseudo-metric which is equivalent to the Teichmuller pseudo-metric was introduced with no use of quasiconformal extensions.
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Research Products
(12 results)
