1992 Fiscal Year Final Research Report Summary
GEOMETRY OF MANIFOLDS AND RELATED SUBJECTS
| Project/Area Number |
03302002
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| Research Category |
Grant-in-Aid for Co-operative Research (A)
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| Allocation Type | Single-year Grants |
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| Research Field |
代数学・幾何学
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| Research Institution | NIIGATA UNIVERSITY |
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Principal Investigator |
SEKIGAWA Kouei Niigata University, Mathematics, Professor, 理学部, 教授 (60018661)
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| Co-Investigator(Kenkyū-buntansha) |
OHMORI Hideki Tokyo Science University, Mathematics, Professor, 理工学部, 教授 (20087018)
SAKANE Yuusuke Osaka University, Mathematics, Associate Professor, 理学部, 助教授 (00089872)
SHIOHAMA Katsuhiro Kyushu University, Mathematics, Professor, 理学部, 教授 (20016059)
NISHIKAWA Seiki Tohoku University, Mathematics, Professor, 理学部, 教授 (60004488)
SUNADA Toshikazu University of Tokyo, Mathematical Sciences, Professor, 数理科学研究科, 教授 (20022741)
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| Project Period (FY) |
1991 – 1992
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| Keywords | Riemannian manifold / Harmonic mapping / Spectrum / Mean curvature / Einstein metric / Gauge theory / Kaehler manifold / Hamiltonian |
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| Research Abstract |
Geometry of manifolds is deeply concerned with many fields of mathematics and furthermore, mathematical physics, technology, information theory, and so on. So, we aim at studying the problems in geometry of manifolds from various view points. In order to achieve our aim, we set the following six research projects, (1) Global analysis, (2) Riemannian geometry, (3) Submanifold theory and Tensor geometry, (4) Geometric structures on manifolds, (5) Complex geometry, (6) Dynamics and dynamical systems on manifolds. We proceeded our research plan by exchanging the progresses in each project to m each other. We here write down some results obtained in our research program. In(1), there are shown some important results concerning the existence and stability of harmonic mappings between noncompact Riemannian manifolds and also obtained concrete construction of harmonic mappings by making use of twistor method. There are some remarkable progresses in the study about topology of 3-and 4-dimensional manifolds by making use of Gauge theory. In (2), there are some progresses in the study of the topics concerning Gromov convergence theorem and also on the structures of noncompact Riemannian manifolds by making use of the geometry of their ideal boundaries. In (3), surfaces with constant mean curvature in a 3-dimensional Euclidean spacer Hyperbolic space are extensively studied. In (4), there is a new trial in the study of geometric treatment of differential equations on manifolds. In (5), there are some progresses in the study concerning the existence and construction of Kaehler-Einstein structures. For example, a counter example to a Yau's conjecture on topological type of Ricci-flat Kaehler manifolds. In (6), for example, a generalization of a theorem by Lioville-Arnold in Hamilton mechanics has been shown. Furthermore, there is a remarkable progress in the study of Statistics from Dynamical view points.
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