2001 Fiscal Year Final Research Report Summary
| Project/Area Number |
11640148
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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 | Tohoku University |
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Principal Investigator |
SHIMIZU Satoru Tohoku University, Graduate School of Science, Associate Professor, 大学院・理学研究科, 助教授 (90178971)
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| Co-Investigator(Kenkyū-buntansha) |
NAKAGAWA Yasuhiro Tohoku University, Graduate School of Science, Lecturer, 大学院・理学研究科, 講師 (90250662)
OGATA Shoetsu Tohoku University, Graduate School of Science, Associate Professor, 大学院・理学研究科, 助教授 (90177113)
KENMOTSU Katsuei Tohoku University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (60004404)
KODAMA Akio Kanazawa University, Faculty of Science, Professor, 理学部, 教授 (20111320)
TAKEUCHI Shigeru Gifu University, Faculty of Education, Professor, 教育学部, 教授 (30021330)
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| Project Period (FY) |
1999 – 2001
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| Keywords | Tube domain / Reinhardt domain / Holomorphic equivalence problem / Holomorphic automorphism group / Lie group / Stability / CR-structure / Constant mean curvature |
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| Research Abstract |
In this research, we have obtained the following results related to special domains.
1. Concerning tube domains, we have established the fundamental result on the prolongation of complete polynomial vector fields on a tube domain, which is called the Prolongation Theorem. Besides, using the Prolongation Theorem, we have obtained a result on the characterization of n-dimensional abelian ideals in the Lie algebra of the affine automorphism group of a tube domain in the n-dimensional complex number space.
2. Related to the study of Reinhardt domains or weakly pseudoconvex domains, we have made a study of the problem of characterizing generalized complex ellipsoids with spherical boundary points, and obtained the Riemann mapping theorem type result that most of such generalized complex ellipsoids coincide with the balls. Moreover, to clarify the aspect that the study of tube domains complements the study of Reinhardt domains, we have tried to give an another proof of this result by making use of the classification of spherical tube manifolds due to Dadok and Yang, and succeeded in the case of a class of generalized complex ellipsoids.
3. Related to the study of the boundaries of special domains, we have investigated the CR structures. In particular, we have clarified a characteristic of set-theoretic representations of DR Lie algebras that correspond to the category dual to CR Lie algebras.
4. As a study of torus actions, we have obtained the result that ample line bundles on nonsingular tone varieties are projectively normal. Also, we have made a detailed study of the two notions of stability for Fano manifolds - the K stability and the CM stability - introduced by Tian related to the Hitchin-Kobayashi correspondence for manifolds.
5. Related to the geometry of the boundaries of special domains, we have given various generalizations of constant mean curvature surfaces.
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Research Products
(13 results)
