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2000 Fiscal Year Final Research Report Summary

Geometric variational problems and submanifolds.

Research Project

Project/Area Number 11640057
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Geometry
Research InstitutionShimane University

Principal Investigator

KIMURA Makoto  Faculty of Science and Engineering, Shimane University, Professor., 総合理工学部, 教授 (30186332)

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)
Project Period (FY) 1999 – 2000
KeywordsMinimal submanifolds / Gauss mapping / circle bundles / Calibrations / Special Lagrangian / Austere submanifolds / Isoparametric hypersurfaces / Ferus' inequality
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.

  • Research Products

    (12 results)

All Other

All Publications (12 results)

  • [Publications] Makoto Kimura: "Minimal immersions of some circle bundle over holomorphic curves in complex quadric to sphere"Osaka J.of Math.. 37・4. 883-903 (2000)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] M.Kimura & S.Maeda: "Geometric meaning of isoparametric hypersurfaces in a real sp."Canad.Math.Bull. 43・1. 74-78 (2000)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] T.Adachi,M.Kimura & S.Maeda: "A characterization of all homogeneous real hypersurfaces in a complex projective space by observing the extrinsic..."Arch.Math.. 73・4. 303-310 (1999)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] V-H.Ki,M.Kimura & S.Maeda.: "Geometry of holomorphic distribution of real hypersurfaces in a complex projective space"Czec.Math.J.. (to appear).

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] T.Adachi,S.Maeda & S.Udagawa: "Simpleness and closedness of cicles in compact Hermatian symmetric space"Tsukuba J.Math.. 24. 1-13 (2000)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] T.Adachi & S.Maeda: "Space forms from the viewpoint of their geodesic spheres"Bull.Austral.Math.Soc.. 62. 205-210 (2000)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Makoto Kimura: "Minimal immersions of some circle bundles over holomorphic curves in complex quadric to sphere"Osaka Math.J.. Vol.37. 883-903 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Makoto Kimura and Sadahiro Maeda: "Geometric meaning of isoparametric hypersurfaces in a real space form"Canad.Math.Bull.. Vol.43. 74-78 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Tosiaki Adachi, Makoto Kimura and Sadahiro Maeda: "A characterization of all homogeneous real hypersurfaces in a complex projective space by observing the extrinsic shape of geodesics"Arch.Math.. Vol.73. 303-310 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] U-Hang Ki, Makoto Kimura and Sadahiro Maeda: "Geometry of holomorphic distributions of real hypersurfaces in a complex projective space"Czec.Math.J.. (to appear.).

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Toshiaki Adachi, Sadahiro Maeda and Seichi Udagawa: "Simpleness and closedness of circles in compact Hermitian symmetric spaces"Tsukuba J.Math.. Vol.24. 1-13 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Toshiaki Adachi and Sadahiro Maeda: "Space forms from the viewpoint of their geodesic spheres"Bull.Austral.Math.Soc.. Vol.62. 205-210 (2000)

    • Description
      「研究成果報告書概要(欧文)」より

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Published: 2002-03-26  

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