Construction of pluriharmonic maps from complex torus into symmetric spaces and applications of the theory of integrable systems
Project/Area Number  09640133 
Research Category 
GrantinAid for Scientific Research (C)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
Geometry

Research Institution  NIHON 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)

Keywords  Harmonic Map / Primitive Map / Compact Symmetric Space / ksymmetric 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 / twotori / 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 nonisotropic harmonic map of 2torus into a sphere or a complex projective space can be lifted to a primitive map of finite type into some ksymmetric space. We generalized the result of F. Burstall to obtain that any weakly conformal nonisotropic harmonic map of 2torus 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 odddimensional sphere. But, this case is included the case of evendimensional 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 nonisotropic 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 ksymmetric space is a composition of a primitive pluriharmonic map of finite type of some Jacobian torus into a ksymmetric space with a Abel map of the compact Riemann surface into the Jacobian torus.

Report
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Research Products
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