KAMISHIMA Yoshinobu Graduate School of Science, Professor, 理学研究科, 教授 (10125304)
OKA Mutsuo Graduate School of Science, Professor, 理学研究科, 教授 (40011697)
OHNITA Yoshihiro Graduate School of Science, Professor, 理学研究科, 教授 (90183764)
INOGUCHI Junichi Fukuoka University School of Science, Research Assistant, 理学部, 助手 (40309886)
UDAGAWA Seiichi Nihon University, School of Medicine, Lecturer, 医学部, 講師 (70193878)
|Budget Amount *help
¥3,500,000 (Direct Cost: ¥3,500,000)
Fiscal Year 2001: ¥1,400,000 (Direct Cost: ¥1,400,000)
Fiscal Year 2000: ¥2,100,000 (Direct Cost: ¥2,100,000)
Results were obtained on the geometry and topology of harmonic maps and spaces of harmonic maps, especially in the case where the domain is a Riemann surface and the target space is a compact Lie group or symmetric space. Guest used a generalization of the Weierstrass representation for minimal surfaces to study harmonic maps from the two-dimensional sphere (or, more generally, harmonic maps of finite uniton number, from any Riemann surface) to the unitary group. Earlier results of Uhlenbeck, Segal, Dorfmeister-Pedit-Wu, Burstall-Guest were developed into an effective tool for describing such maps. In particular, an explicit canonical form was obtained, and this was used to study the space of all such maps. The main application was a description of the connected components of the space of harmonic maps from the two-dimensional sphere to the unitary group. Ohnita used a different approach, based on earlier work of Hitchin in gauge theory, to obtain a framework for studying the geometry (in particular, the pre-symplectic geometry) of spaces of harmonic maps.
The harmonic map equation can be regarded as an integrable system, and the above work sheds light on other integrable systems. Two other examples of integrable systems were studied from this point of view, and preliminary results obtained. The first example, studied by Guest, was the theory of quantum differential equations. Parallels with harmonic maps were established, forming the basis for future work in this direction. Results on quantum cohomology of symmetric spaces were obtained also by Ohnita and Nishimori, and on quantum cohomology of flag manifolds by Guest and Otofuji. The second example, studied by Burstall and Calderbank, was the integrable systems aspect of conformal and Mobius geometry, and a new approach was initiated.