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

本课题在研究期间进行了大量的研究工作，取得了以下丰硕的研究成果。Ohnita和Udagawa关于有限型调和映射的联合工作发表在第9届MSJ-IRI会议上。它涉及到不同k-对称空间上的扭圈代数之间的等价性问题，我们将进一步研究。Ohnita从可积系统的观点讨论了对称空间中的多重调和映射，并证明了多重调和映射的DPW公式。在联合工作与詹姆斯Eells的结构空间的调和映射我们开始从精确的证明，空间的调和映射之间的紧凑真实的解析黎曼流形是一个真实的解析空间，我们仍在工作。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Lagrange子流形的Hamilton稳定性问题。通过李理论的方法，他证明了嵌入复射影空间中的紧致极小不可约对称拉格朗日子流形是Hamilton稳定的。此外，我们还证明了紧致对称拉格朗日子流形嵌入复欧氏空间。并讨论了拉格朗日子流形与矩映射的关系。到目前为止，只知道复射影空间和复欧氏空间中的Hamilton稳定拉格朗日子流形。是真实的射影子空间和Clifford环面。而在具有平行第二基本形式的拉格朗日子流形类中，我们给出了许多Hamilton稳定的拉格朗日子流形，即对称拉格朗日子流形的丰富例子。小池在非紧型的情形下成功地建立了Hilbert空间中的复等焦子流形和等参子流形的理论。这是对Terng-Thorgergsson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.