Differential geometry of harmonic maps, minimal submanifolds and Yang-Mills-Higgs equations

在这个项目中，我们在研究期间进行了大量的研究活动，并获得了以下丰硕的研究成果。我们期待着更好的研究进展。Ohnita和Udagawa关于有限型谐波映射的联合研究发表在第9届日本国际物理学会会刊上。它涉及到不同k对称空间的扭环代数间的等价问题，我们将进一步研究。Ohnita从可积系统的角度讨论了对称空间中的多谐映射，并证明了多谐映射的DPW公式。在与James Eells合作的关于调和映射空间结构的工作中，我们从精确证明紧实解析黎曼流形之间的调和映射空间是一个实解析空间开始，我们还在继续工作。Ohnita从极小子流形理论的一个新领域出发，研究了K"ahler流形中拉格朗日子流形的哈密顿稳定性问题。利用李论方法，证明了嵌入复投影空间中的紧极小不可约对称拉格朗日子流形是哈密顿稳定的。此外，我们证明了紧对称拉格朗日子流形嵌入复欧几里德空间。讨论了拉格朗日子流形与矩映射之间的关系。迄今为止只知道复投影空间和复欧几里德空间中的哈密顿稳定拉格朗日子流形。是实数射影子空间和Clifford环面。然而，我们给出了具有平行第二基本形式的拉格朗日子流形，即对称拉格朗日子流形中哈密顿稳定拉格朗日子流形的许多丰富的例子。小池成功地构造了对称空间中的复等焦子流形和非紧型情况下Hilbert空间中的等参子流形的理论。这是对滕-托格森提出的一个问题的回答。

In this project we had much research activity during the research period and we obtained the following fruitful research results. We expect fitter research progress.The joint work of Ohnita and Udagawa on harmonic maps of finite type was published in the proceedings of the 9-th MSJ-IRI. It is related with the equivalence problem among twisted loop algebras associated with different k-symmetric spaces and we will go to further research. And Ohnita discussed pluriharmonic maps into symmetric spaces from the viewpoint of integrable systems and proved DPW formula for pluriharmonic maps. In the joint work with James Eells on the structure of spaces of harmonic maps we started from the precise proof that the space of harmonic maps between compact real analytic Riemannian manifols is a real analytic space, and we are still working. From the viewpoint of a new area in minimal submanifold theory, Ohnita studies the Hamiltonian stability problem of Lagrangian submanifolds in K"ahler manifolds. By the Lie theoretic method, he showed that compact minimal irreducible symmetric Lagrangian submanifolds embedded in complex projective spaces are Hamiltonian stable. Moreover, we proved that compact symmetric Lagrangian submanifolds embedded in complex Euclidean spaces. And we discuss the relationship between Lagrangian submanifolds and the moment maps. Until now only known Hamiltonian stable Lagrangian submanifolds in complex projective spaces and complex Euclidean spaces. Were real projective subspaces and Clifford tori. However we gave many rich examples of Hamiltonian stable Lagrangian submanifolds in the class of Lagrangian submanifolds with parallel second fundamental form, namely symmetric Lagrangian submanifolds. Koike has succeeded in construction of theory for complex equifocal submanifolds in symmetri spaces and isoparametric submanifolds in Hilbert spaces in the case of noncompact type. It is an answer to a problem posed by Terng-Thorgergsson.