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

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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.