Geometry of the flat tori in the sphere and non- linear wave equations

在这项研究中，我们研究了平面环面在三维球面、亚纯映射、常平均曲率曲面和动力系统中的几何。本文的主要研究成果如下：1.研究了三维球面上的平坦环面。在这项研究中，Y.Kitagawa研究了$S^3$2中紧致曲面的任何等距变形保持封闭体积的猜想，从而证明了该猜想对$S^3$2中的所有平坦环面都成立。在这项研究中，Y.Aihara证明了对于复射影空间中每个次数为$d的超曲面$D，都存在一条从复平面到射影空间的全纯曲线，它的亏量小于1。3.常平均曲率曲面和Backlund变换的研究。在这项研究中，J.Inoguchi证明了常平均曲率曲面的Bianci-Backlund变换与Darboux变换和简单类型修整是等价的。4.动力系统的研究。在这项研究中，K.Sakai证明了流形上扩张向量场集的$C^1内被刻画为同时满足公理A和拟横截条件的无奇点向量场集。

In this research, we studied geometry of flat tori in the 3-sphere, meromorphic mappings, surfaces of constant mean curvature and dynamical systems. The main results of this reseach are summarized as follows.1.Studies on flat tori in the 3-sphere. In this research, Y.Kitagawa studied the conjecture that any isometric deformation of compact surface in $S^3$ preserves the enclosed volume.As a result, he proved that the conjecture is ture for all flat tori in $S^3$.2.Studies on meromorphic mappings. In this research, Y.Aihara proved that for every hypersurface $D$ of degree $d$ in a complex projective space, there exists a holomorphic curve from the complex plane into the projective space whose deficiency for $D$ is positive and less than one.3.Studies on constant mean curvature surfaces and Backlund transformations. In this research, J.Inoguchi proved that Bianchi-Backlund transformation of a constant mean curvature surface is equivalent to the Darboux transformation and the simple type dressing.4.Studies on dynamical systems. In this research, K. Sakai proved that the $C^1$ interior of the set of expansive vector fields on a manifold is characterized as the set of vector fields without singularities satisfying both Axiom A and the quasi-transversality condition.