1999 Fiscal Year Final Research Report Summary
Complex Manifolds and Gauge Theory
| Project/Area Number |
09440027
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Geometry
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| Research Institution | TOHOKU UNIVERSITY |
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Principal Investigator |
BANDO Shigetoshi Graduate School of Science, Tohoku Univ., Prof., 大学院・理学研究科, 教授 (40165064)
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| Co-Investigator(Kenkyū-buntansha) |
ISHIDA Masanori Graduate School of Science, Tohoku Univ., Prof., 大学院・理学研究科, 教授 (30124548)
URAKAWA Hajime Graduate School of Information Science, Tohoku Univ., Prof., 大学院・情報科学研究科, 教授 (50022679)
NISHIKAWA Seiki Graduate School of Science, Tohoku Univ., Prof., 大学院・理学研究科, 教授 (60004488)
IZEKI Hiroyasu Graduate School of Science, Tohoku Univ., Assoc. Prof., 大学院・理学研究科, 助教授 (90244409)
TAKAGI Izumi Graduate School of Science, Tohoku Univ., Prof., 大学院・理学研究科, 教授 (40154744)
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| Project Period (FY) |
1997 – 1999
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| Keywords | Einstein / harmonic maps / Carnot groups / Laplace operators on graphs / toric varieties / reaction-diffusion system / complex Kleinian groups / Futaki character |
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| Research Abstract |
Bando studied the existence problems of Einstein metrics on Kahler manifolds and holomorphic complex vector bundles. It is believed that there must be good relations between the existence of Einstein metrics and stabilities. He obtained a useful formula on a functional which connects them. He also wrote a paper which shows how Green functions can be used to obtain harmonic geometric objects.
Nishikawa, jointly with Keisuke Ueno (Yamagata Univ.), studied the Dirichlet problem at infinity for harmonic maps between homogeneous spaces of negative curvature, and the complex analyticity of harmonic maps between complex hyperbolic spaces. A proper harmonic map which is CィイD14ィエD1 upto boundary and gives non-degenerate CR map on the boundaries is shown to be holomorphic.
Urakawa continued to study harmonic maps, Yang-Mills connections and etc., and generalized the methods to work on graphs. On finite or infinite graphs, he obtained results on the spectra of Laplace operators, the estimates on Gr
… More
een functions and the analog of harmonic maps.
Ishida studied real fans which generalize (rational) fans which are closely related to toric varieties. He introduced a category of graded modules of exterior algebra over real fans and defined a dualizing functor. He obtained counterparts of Serre duality and Poincare duality.
Takagi studied a reaction-diffusion system which is posed by A. Gierer and H. Meinhardt as a fundamental model of morphogenesis and a constrained variational problem on a bending functional which gives a model of the shape transformation of erythrocyte.
Izeki studied entoropy rigidity and convex compactness of Kleinian groups acting on real space forms. He obtained a partial resolution of a conjectire on the inequality between the Hausdorff dimension of the limit sets of convex co-compact Kleinian groups and the cohomological dimension of the groups.
Nakagawa studied Bando-Calabi-Futaki characters and generalized some of properties which were known to Fano manifolds to general projective manifolds and their Kahler classes. Under certain assumption, he showed a vanishing of Bando-Calabi-Futaki characters on the Lie algebra of unipotent groups and the existence of lifts of Bando-Calabi-Futaki characters to group characters. Less
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