1991 Fiscal Year Final Research Report Summary
Stability and complex-analyticity of pluriharmonic maps from compact Kahler manifolds, and their constructions and factorizations.
| Project/Area Number |
01540078
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| Research Category |
Grant-in-Aid for General Scientific Research (C)
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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 | Nihon University |
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Principal Investigator |
IKAWA Toshihiko Nihon University, School of Medicine, Associate Professor, 医学部, 助教授 (30151252)
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| Co-Investigator(Kenkyū-buntansha) |
UDAGAWA Seiichi Nihon University, School of Medicine, Lecturer, 医学部, 講師 (70193878)
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| Project Period (FY) |
1989 – 1991
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| Keywords | Pluriharmonic map / Holomorphic map / Complex manifold / Complex Grassmann manifold / Stability / Complex-analyticity / Slant immersion |
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| Research Abstract |
In the first year of this research, we investigated the stability and complex-analyticity of pluriharmonic maps from compact Kahler manifolds into irreducible Hermitian symmetric spaces of compact type. As a result, we proved that any stable harmonic map from complex projective space with Fubini-Study metric into compact Kahler manifold of positive holomorphic bisectional carvatue is holomorphic or anti-holomorphic. We also got the fundamental tools of constructing pluriharmonic maps explicitly.
In the second year of this research, we tried to solve the conjecture that any pluriharmonic map from compact complex manifold into complex Grassmann manifold is constructed explicitly from a holomorphic map. This conjecture, of course, depends on the fact that any harmonic map from Riemann spher into complex Grassmann manifold is constructed explicitly from a holomorphic map, which is a result due to Eells' school and Chern-Wolfson. Note that in case the domain is Riemann surface, the concept of pluriharmonic maps coincides with that of harmonic maps. Now, it is clear that we must restrict the domain manifold to have positive first chern class. As a result, any pluriharmonic map from compact complex manifold with positive first chern class into complex Grassmann manifold G_k(*^n) is constructed explicitly from a holomorphic map if, k=1 or 2, or, k=3 and n*14, or k=4 and n*15. In the last year of this research, we proved that the above examples of pluriharmonic maps give the examples of the slant immersions, which is a notion B. Y. Chern introduced, if they are equivariant isometric immersion. In general, we proved that equivariant isometric immersion from Kahler *-space with b_2=1 into complex projective space is a slant immersion.
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