2006 Fiscal Year Final Research Report Summary
Study on the theory of conformal embeddings of a Riemann surface focused on the hyperrbolic metric and hydrodynamics of viscous fluids
| Project/Area Number |
16540157
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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 | Hiroshima University |
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Principal Investigator |
SHIBA Masakazu Hiroshima University, Graduate School of Engineering, Professor (70025469)
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| Co-Investigator(Kenkyū-buntansha) |
MIZUTA Yoshihiro Hiroshima University, Graduate School of Science, Prof. (00093815)
AMANO Kaname Ehime Univ., Faculty of Eng., Prof. (80113512)
MASUMOTO Makoto Yamaguchi Univ., Grad. School of of Sci. and Eng., Prof. (50173761)
MAITANI Fumio Kyoto Insti. of Tech., Grad. School of Sci. and Tech., Prof. (10029340)
ITO Masaaki Hiroshima University, Graduate School of Engineering, Assoc. Prof. (10116535)
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| Project Period (FY) |
2004 – 2006
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| Keywords | Riemann surfaces / Hyperbolic metric / Conformal embedding s / Viscous fluid flow s / Poiseuille flow / Univalent functions / Discontinuous groups s / Extremal parallel slit mappings |
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| Research Abstract |
Our main purpose was to study conformal embeddings of a Riemann surface into another and hydrodynamics of viscous fluids. Concerning the first topic, we obtained the notion of "circularizable domains" on a Riemann surface and showed that these domains play an important role in the proof of classical Riemann mapping theorem and give a new tool in the construction of fundamental domains for a discontinuous group of conformal automorphisms … such as Fuchsian groups. The second topic is based on the insight that the various area theorems in the theory of conformal embedding will be closely connected with the viscous fluid flows in a tube. The conjecture was first verified in the fall of 2006. We have succeeded in determining the viscosity of the fluid in a tube which realizes the absolute area theorem due to the head investigator. We have thus obtained generalization of the Poiseuille flows. The results have been announced in an international meeting held in Germany and also in the Fall Meeting, Mathematical Society of Japan. Besides, as an application of the theory of conformal embedding, we determined a necessary and sufficient condition for a complex linear combination of extremal parallel slit mappings, which gives a solution to a problem of Maitani.
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