2000 Fiscal Year Final Research Report Summary
Geometric variational problems and submanifolds.
| Project/Area Number |
11640057
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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 |
Geometry
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| Research Institution | Shimane University |
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Principal Investigator |
KIMURA Makoto Faculty of Science and Engineering, Shimane University, Professor., 総合理工学部, 教授 (30186332)
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| Co-Investigator(Kenkyū-buntansha) |
HATTORI Yasunao Faculty of Science and Engineering, Shimane University, Professor., 総合理工学部, 教授 (20144553)
MAEDA Sadahiro Faculty of Science and Engineering, Shimane University, Professor., 総合理工学部, 教授 (40181581)
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| Project Period (FY) |
1999 – 2000
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| Keywords | Minimal submanifolds / Gauss mapping / circle bundles / Calibrations / Special Lagrangian / Austere submanifolds / Isoparametric hypersurfaces / Ferus' inequality |
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| Research Abstract |
First we invetigated 3-dimensional minimal submanifolds with 2-parameter family of great spheres in a sphere S^n. Set of (oriened) great circles is identified with real (oriented) 2-plane Grassmannian and the complex quadric Q^<n-1> in a complex projective space. Then the submanifold M with 2-parameter family of great spheres in S^n is constructed as a circle bundle over a 2-dimensional surface Σ in Q^<n-1>. We showed that (1) Σ is a complex 1-dimensional holomorphic curve in Q^<n-1>, then the Gauss mapping of the corresponding submanifold M in S^n is degenerate, (2) the holomorphic curve Σ in Q^<n-1> is first order isotropic, then the corresponding M is minimal.
Next, by a joint research with Goo Ishikawa (Hokkaido Univ.) and Reiko Miyaoka (Sophia Univ.), we generalized the former results to higher dimensional submanifolds in spheres. Especially, if a complex submanifold Σ in Q^<n-1> is first order isotropic, then the corresponding submnanifold M (circle bundle over Σ) with (dim_R Σ)-parameter family of great spheres in S^n is austere. Hence we can construct special Lagrangian submanifolds in complex Euclidean spaces by using the results with respect to the calibration by Harvey and Lawson. And we showed that from some homogeneous submanifolds in real Grassmannians of rank 2, 3, 5, one can construct homogeneous austere submanifolds M in S^n such that the Gauss mapping of M is degenerate and satisfying Ferus' equality. They are a natural generalization of E.Cartan's isoparametric hypersurfaces.
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Research Products
(12 results)
