HARMONIC MAPS INTO SYMMETRIC SPACES AND GEOMETRY OF MODULI SPACES

到对称空间的调和映射具有到一般黎曼流形的调和映射所不具有的几个特征性质。例如，这种调和映射方程可以表示为零曲率方程、Lax方程和规范理论方程。作为研究调和映射到对称空间的途径，我们研究了与调和映射相关的规范论方程及其解的模空间的结构和几何，得到了几个结果.我已经写了题为“几何模空间的调和映射到李群通过规范理论在黎曼曲面”这一工作是由外国研究人员估计，因为它是非常有趣和启发性的文件。进一步研究了紧对称空间中的一类调和映射--有限型调和映射。引入了从紧Riemann曲面到紧k-对称空间的广义有限型调和映射的概念， 关于我们 证明了这样的调和映射是紧致Riemann曲面到Jacobi簇的Abel映射和Jacobi簇到k-对称空间的多重调和映射的合成.我们已经写了题为“调和映射有限型到广义旗流形和扭量纤维”的文件，他们将发表在国际。数学和伦敦数学学会，分别另一方面，作为与可积系统相关的研究，我们关注Frobenius流形与多重调和映射之间的关系。利用对称空间理论构造了复射影空间和Hermitian对称空间中的紧致极小Lagrangian子流形，并研究了其Hamilton稳定性问题，得到了新的结果.讨论了对称R-空间在对称空间中的良好性质，并给出了对称R-空间的新刻画。在德国波恩的Max-Planck-Institut fuer Mathematik等的停留期间，从对称空间范畴的基本理论的观点对奇维欧氏空间的完备迷向极小曲面的性质和构造进行了研究。合作者Hashiguchi Hideko在这次研究会上做了关于从黎曼球面到酉群的调和映射对应的酉的模空间问题的研究报告。少

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 gauge-theoretic equations. As an approach for harmonic maps into symmeric spaces, we investigate the gauge-theoretic 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 k-symmetric spac … More 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 k-symmetric 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 R-spaces with nice properties in symmetric spaces and gave the new characterization of symmetric R-spaces from the viewpoint of the basic theory in the category of symmetric spaces through stays at Max-Planck-Institut fuer Mathematik in Bonn, Germany etc. The collaborator, Masatoshi Kokubu,showed new results on propeties and construction of complete isotropic minimal surfaces in odd-dimensional 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