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Stability and complex-analyticity of pluriharmonic maps from compact Kahler manifolds, and their constructions and factorizations.

Research Project

Project/Area Number 01540078
Research Category

Grant-in-Aid for General Scientific Research (C)

Allocation TypeSingle-year Grants
Research Field 代数学・幾何学
Research InstitutionNihon University

Principal Investigator

IKAWA Toshihiko (1990-1991)  Nihon University, School of Medicine, Associate Professor, 医学部, 助教授 (30151252)

宇田川 誠一 (1989)  日本大学, 医学部, 講師 (70193878)

Co-Investigator(Kenkyū-buntansha) UDAGAWA Seiichi  Nihon University, School of Medicine, Lecturer, 医学部, 講師 (70193878)
井川 俊彦  日本大学, 医学部, 助教授 (30151252)
Project Period (FY) 1989 – 1991
Project Status Completed (Fiscal Year 1991)
Budget Amount *help
¥1,500,000 (Direct Cost: ¥1,500,000)
Fiscal Year 1991: ¥500,000 (Direct Cost: ¥500,000)
Fiscal Year 1990: ¥500,000 (Direct Cost: ¥500,000)
Fiscal Year 1989: ¥500,000 (Direct Cost: ¥500,000)
KeywordsPluriharmonic map / Holomorphic map / Complex manifold / Complex Grassmann manifold / Stability / Complex-analyticity / Slant immersion / Pluriharwonic Wap / Slant immeusion / Kahler CーSpace / Complex Projective space / Complex Grossmann manifold / plariharmonic map / holomorphic map / complex manifold / complex Grassmann manifold / explicit construction / positive first Chern class / pluriharmonic map / holoworphic map / rational map / complex manitold / complex Grasswann manifold / stability / compex-analyticity
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.

Report

(4 results)
  • 1991 Annual Research Report   Final Research Report Summary
  • 1990 Annual Research Report
  • 1989 Annual Research Report
  • Research Products

    (22 results)

All Other

All Publications (22 results)

  • [Publications] 大仁田 義裕,宇田川 誠一: "Complexーanalyticity of pluriharmonic maps and their constructions" “Prospects in Complex Geametry"(Proceedings,katatal Kyoto 1989)edited by J.Noguchi and T.Ohsawa,Lect.Notos in Math.,SpriugerーVerlag. 1468. 371-407 (1991)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1991 Final Research Report Summary
  • [Publications] 大仁田 義裕,宇田川 誠一: "Stability,complexーanalyticity and constancy of plurihavmonic maps from compact Kahler manifolds" Mathematische Zeitschrift. 205. 629-644 (1990)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1991 Final Research Report Summary
  • [Publications] 井川 俊彦,宇田川 誠一: "Immevsious of complex projective spaces into complex projective spaces" 日本大学医学部進学課程研究紀要. 17. 15-27 (1989)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1991 Final Research Report Summary
  • [Publications] 宇田川 誠一: "A explicit method of constructing pluriharmonic maps from compact complex manifold into complex Grassmann manifold"

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1991 Final Research Report Summary
  • [Publications] 前田 定廣,大仁田 義裕,宇田川 誠一: "On Slant immersions into Kahler manifolds"

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1991 Final Research Report Summary
  • [Publications] Y. Ohnita and S. Udagawa: ""Complex-analyticity of pluriharmonic maps and their constructions"" Lect. Notes in Math., Springer-Verlag. No. 1468. 371-407 (1991)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1991 Final Research Report Summary
  • [Publications] Y. Ohnita and S. Udagawa: ""Stability, complex-analyticity and constancy of pluriharmonic maps from compact Kahler manifolds"" Math. Zeitschrift. No. 205. 629-644 (1990)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1991 Final Research Report Summary
  • [Publications] T. Ikawa and S. Udagawa: ""Immersions of complex projective spaces into complex projective spaces"" No. 17. 15-27 (1989)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1991 Final Research Report Summary
  • [Publications] S. Udagawa: ""A explicit method of constructing pluriharmonic maps from compact complex manifold into complex Grassmann manifold""

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1991 Final Research Report Summary
  • [Publications] S. Maeda, Y. Ohnita and S. Udagawa: ""On slant immersions into Kahler manifolds""

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1991 Final Research Report Summary
  • [Publications] 大仁田 義裕,宇田川 誠一: "Complexーanalyticity of pluriharmonic waps and their constructions" “Prospects in Complex Geometry"(Proceedings,Katata/Kyoto 1989),edited by J.Moguchi and T.Ohsawa,Lect.Notes in Math.,SprinerーUerlag. 1468. 371-407 (1991)

    • Related Report
      1991 Annual Research Report
  • [Publications] 大仁田 義裕,宇田川 誠一: "Stabilty,Complexーanalyticity and constancy of pluriharmonic waps from compact kahler wanifolds" Mathewatisde Zeitschrift. 205. 629-644 (1990)

    • Related Report
      1991 Annual Research Report
  • [Publications] 井川 俊彦,宇田川 誠一: "Immersions of complex projective spaces into complex profective spaces" 日本大学医学部進学課程研究紀要. 17. 15-27 (1989)

    • Related Report
      1991 Annual Research Report
  • [Publications] 宇田川 誠一: "A explicit method of constructing pluriharwonic waps from compact complex manifold into complex Grassmann manifold"

    • Related Report
      1991 Annual Research Report
  • [Publications] 前田 定廣,大仁田 義裕,宇田川 誠一: "On slant immerstions into Kahler wanifolds"

    • Related Report
      1991 Annual Research Report
  • [Publications] 大仁田 義裕、宇田川 誠一: "Complexーanalyticity of pluriharmonic maps and their constructions" Lecture Notes in Math. (SpringerーVerlag).

    • Related Report
      1990 Annual Research Report
  • [Publications] 大仁田 義裕,宇田川 誠一: "Stability,complexーanalyticity and constancy of pluriharmonic maps from compact Kahli manifolds" Math、 Zeit.,.

    • Related Report
      1990 Annual Research Report
  • [Publications] 井川 俊彦 宇田川 誠一: "Immersions of complex projective spaces into complex projective spaces" 日本大学医学部進学課程研究紀要. 17. 15-27 (1989)

    • Related Report
      1990 Annual Research Report
  • [Publications] 宇田川 誠一: "A explicit method of constructing pluriharmonic maps from compact complex manifold into complex Grassman manifold" preprint.

    • Related Report
      1990 Annual Research Report
  • [Publications] 大仁田義裕,宇田川誠一: "Complex-analyticity of pluriharmonic maps and their constructions" Lecture Notes in Mathematics,Springer-Verlag.

    • Related Report
      1989 Annual Research Report
  • [Publications] 大仁田義裕,宇田川誠一: "Stability,complex-analyticity and constancy of plurihanuonic maps from compact kahler manifolds" Math.Zeit.

    • Related Report
      1989 Annual Research Report
  • [Publications] 井川俊彦,宇田川誠一: "Immersions of complex projective spaces into complex projective spaces" 日本大学医学部進学課程研究紀要第17号. 17. (1989)

    • Related Report
      1989 Annual Research Report

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Published: 1989-04-01   Modified: 2016-04-21  

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