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Construction of pluriharmonic maps from complex torus into symmetric spaces and applications of the theory of integrable systems

Research Project

Project/Area Number 09640133
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Geometry
Research InstitutionNIHON UNIVERSITY

Principal Investigator

UDAGAWA Seiichi  School of Medicine, NIHON UNIVERSITY, Lecturer, 医学部, 講師 (70193878)

Project Period (FY) 1997 – 1999
Project Status Completed (Fiscal Year 1999)
Budget Amount *help
¥1,500,000 (Direct Cost: ¥1,500,000)
Fiscal Year 1999: ¥500,000 (Direct Cost: ¥500,000)
Fiscal Year 1998: ¥500,000 (Direct Cost: ¥500,000)
Fiscal Year 1997: ¥500,000 (Direct Cost: ¥500,000)
KeywordsHarmonic Map / Primitive Map / Compact Symmetric Space / k-symmetric space / finite type / twistor fibration / Pluriharmonic Map / Abel Map / ツィスター射影 / ツィスター構成 / Harmonic map / two-tori / complex Grass mannian / Spectral carve / dressing transtorm / トーラス / 複素グラスマン多様体 / スペクトル・データ / 非超極小
Research Abstract

In our study, I obtained three main results. I expain each of them separately.
(1) F. Burstall proved that any weakly conformal non-isotropic harmonic map of 2-torus into a sphere or a complex projective space can be lifted to a primitive map of finite type into some k-symmetric space. We generalized the result of F. Burstall to obtain that any weakly conformal non-isotropic harmonic map of 2-torus into a sphere or a complex projective space is itself of finite type. In fact, we can prove a more general result. Given a primitive map of finite type into a generalized flag manifold, we can project it into some compact symmetric space as a harmonic map of finite type under some condition on the choice of isotropy subgroup of the compact symmetric space. The condition is rather mild and satisfied by the above cases (except odd-dimensional sphere. But, this case is included the case of even-dimensional sphere).
(2) We extended the result (1) to the case where the domain is a complex manifold and pluriharmonic maps into it. We proved that any non-isotropic pluriharmonic map of complex torus into a complex projective space is of finite type.
(3) We introduced the concept of primitive maps of generalized finite type and obtained some results on the harmonic maps of compact Riemann surfaces of higher genus. In fact, any primitive harmonic maps of generalized finite type of a compact Riemann surface of genus greater than 1 into a k-symmetric space is a composition of a primitive pluriharmonic map of finite type of some Jacobian torus into a k-symmetric space with a Abel map of the compact Riemann surface into the Jacobian torus.

Report

(4 results)
  • 1999 Annual Research Report   Final Research Report Summary
  • 1998 Annual Research Report
  • 1997 Annual Research Report
  • Research Products

    (4 results)

All Other

All Publications (4 results)

  • [Publications] 大仁田義裕: "球面およびトーラス面から対称空間への調和写像の分類問題"京都大学数理解析研究所講究録. 1113. 44-64 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Y. Ohnita: "Classification problem of harmonic maps from Riemann spheres and two-torus into symmetric space"RIMS, Koukyuuroku. 1113. 44-64 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] 大仁田義裕: "球面およびトーラス面から対称空間への調和写像の分類問題(解説と未解決問題)"京都大学数理解析研究所講究録. 1113. 44-64 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] S.Udagawa: "An exposition of McIntosh's work by showing many examples" 日本大学医学部一般教育研究紀要. 26号. 13-33 (1998)

    • Related Report
      1998 Annual Research Report

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Published: 1997-04-01   Modified: 2016-04-21  

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