2003 Fiscal Year Final Research Report Summary
Differential geometry of harmonic maps, minimal submanifolds and Yang-Mills-Higgs equations
| Project/Area Number |
13440025
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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 | Tokyo Metropolitan University |
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Principal Investigator |
OHNITA Yoshihiro Tokyo Metropolitan University, Department of Mathematics, Professor, 大学院・理学研究科, 教授 (90183764)
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| Co-Investigator(Kenkyū-buntansha) |
MARTIN Guest Tokyo Metropolitan University, Department of Mathematics, Professor, 大学院・理学研究科, 教授 (10295470)
MIYAOKA Reiko Kyushu University, Graduate School of Mathematics, Professor, 教理学研究院, 教授 (70108182)
KOIKE Naoyuki Tokyo University of Science, Department of Mathematics, Lecturer, 理学部, 講師 (00281410)
UDAGAWA Seiichi Nihon University, Department of Mathematics, Associate professor, 医学部, 助教授 (70193878)
MORIYA Katsuhiro University of Tsukuba, Department of Mathematics, Research associate, 数学系, 助手 (50322011)
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| Project Period (FY) |
2001 – 2003
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| Keywords | differential geometry / harmonic map / minimal submanifold / Yang-Mills-Higgs equation / moduli space / Lagrange submanifold |
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| Research Abstract |
In this project we had much research activity during the research period and we obtained the following fruitful research results. We expect fitter research progress.
The joint work of Ohnita and Udagawa on harmonic maps of finite type was published in the proceedings of the 9-th MSJ-IRI. It is related with the equivalence problem among twisted loop algebras associated with different k-symmetric spaces and we will go to further research. And Ohnita discussed pluriharmonic maps into symmetric spaces from the viewpoint of integrable systems and proved DPW formula for pluriharmonic maps. In the joint work with James Eells on the structure of spaces of harmonic maps we started from the precise proof that the space of harmonic maps between compact real analytic Riemannian manifols is a real analytic space, and we are still working. From the viewpoint of a new area in minimal submanifold theory, Ohnita studies the Hamiltonian stability problem of Lagrangian submanifolds in K"ahler manifolds. By the Lie theoretic method, he showed that compact minimal irreducible symmetric Lagrangian submanifolds embedded in complex projective spaces are Hamiltonian stable. Moreover, we proved that compact symmetric Lagrangian submanifolds embedded in complex Euclidean spaces. And we discuss the relationship between Lagrangian submanifolds and the moment maps. Until now only known Hamiltonian stable Lagrangian submanifolds in complex projective spaces and complex Euclidean spaces. Were real projective subspaces and Clifford tori. However we gave many rich examples of Hamiltonian stable Lagrangian submanifolds in the class of Lagrangian submanifolds with parallel second fundamental form, namely symmetric Lagrangian submanifolds. Koike has succeeded in construction of theory for complex equifocal submanifolds in symmetri spaces and isoparametric submanifolds in Hilbert spaces in the case of noncompact type. It is an answer to a problem posed by Terng-Thorgergsson.
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