2001 Fiscal Year Final Research Report Summary
HARMONIC MAPS INTO SYMMETRIC SPACES AND GEOMETRY OF MODULI SPACES
Project/Area Number 
11640088

Research Category 
GrantinAid for Scientific Research (C)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
Geometry

Research Institution  Tokyo Metropolitan University 
Principal Investigator 
OHNITA Yoshihiro GRADUATE SCHOOL OF SCIENCE, DEPARTMENT OF MATHEMATICS, TOKYO METOROPOLITAN UNIVERSITY PROFESSROR > 東京都立大学, 理学(系)研究科(研究院), 教授 (90183764)

CoInvestigator(Kenkyūbuntansha) 
NAGATOMO Yasuyuki DEPARTMENT OF MATHEMATICS, UNIVERSITY OF TSUKUBA, LECTURER, 数学系, 講師 (10266075)
KAMISHIMA Yoshinobu FACULTY OF SCIENCE, DEPARTMENT OF MATHEMATICS, TOKYO METOROPOLITAN UNIVERSITY PROFESSROR, 理学(系)研究科(研究院), 教授 (10125304)
GUEST Martin A. GRADUATE SCHOOL OF SCIENCE, DEPARTMENT OF MATHEMATICS, TOKYO METOROPOLITAN UNIVERSITY PROFESSROR, 理学(系)研究科(研究院), 教授 (10295470)
KOKUBU Masatoshi DEPARTMENT OF NATURAL SCIENCES, FACULTY OF ENGINEERING, TOKYO DENKI UNIVERSITY, LECTURER, 工学部, 講師 (50287439)
TANAKA Makiko DEPARTMENT OF MATHEMATICS, SCIENCE UNIVERSITY OF TOKYO, LECTURER, 理工学部, 講師 (20255623)

Project Period (FY) 
1999 – 2000

Keywords  Harmonic map / Symmetric space / Moduli space / YangMillsHiggs equation / Integrable system 
Research Abstract 
Harmonic maps into symmetric spaces have several characteristic properties that harmonic maps into general Riemannian manifolds do not have. For example such harmonic map equation can be formulated as the zero curvature equation, the Lax equation and gaugetheoretic equations. As an approach for harmonic maps into symmeric spaces, we investigate the gaugetheoretic equations associated to such harmonic maps and the structure and the geometry of the moduli spaces of their solutions, and we obatined several results. I have written up the paper entitled with "Geoemtry of the moduli spaces of harmonic maps into Lie groups via gauge theory over Riemann surfaces" This work was estimated by foreign researchers as it is very interesting and imformative. Furthermore, we studied harmonic maps of finite type which is a class of harmonic maps into compact symmetric spaces. We introduced the notion of harmonic maps of generalized finite type from compact Riemann surfaces to compact ksymmetric spac
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es, and we proved that such a harmonic map is the composition of the Abel map from a compact Riemann surface to the Jacobi variety and a pluriharmonic map from the Jacobi variety to a ksymmetric space. We have written up the paper entitled with "Harmonic maps of finite type into generalized flag manifolds and twistor fibrations" They will be published in Inter. J. Math. And J. London Math. Soc., respectively. On the other hand, as the research related to integrable systems, we give our attention to the relationship between Frobenius manifolds and pluriharmonic maps. It is now in progress to study Hamiltonian stability problem for compact minimal Lagrangian submanifolds in complex projective spaces and Hermitian symmetric spaces constructed by using the symmetric space theory and we obtain new results on it. The collaborator, Makiko Tanaka, treated symmetric Rspaces with nice properties in symmetric spaces and gave the new characterization of symmetric Rspaces from the viewpoint of the basic theory in the category of symmetric spaces through stays at MaxPlanckInstitut fuer Mathematik in Bonn, Germany etc. The collaborator, Masatoshi Kokubu,showed new results on propeties and construction of complete isotropic minimal surfaces in odddimensional Euclidean space. The collaborator, Hideko Hashiguchi, gave a research report on problems about moduli space of unitons corresponding to harmonic maps from a Riemann sphere into the unitary group at this research meeting. Less

Research Products
(13 results)