Surfaces with parallel mean curvature vector in Complex projective spaces

本文的研究目的是利用各种方法从多个角度研究复射影空间的性质。每位研究者的目标是利用自己的专业知识开展研究，并取得以下研究成果。本课题组组长Ogata研究了复射影空间中具有平行平均曲率向量场的曲面的表征和分类，并证明了二维复射影空间中存在具有恒定Kaehler角和非恒定高斯曲率的极小曲面。Hirabuki用半群环产生环的证明定理的方法证明了半群的Krull-Akizuki定理。利用几何技术在四元数射影空间的切束上构造了复杂结构，并证明了这些复杂结构与Furuya-Tanaka-Yoshizawa给出的结构一致。Sawada从代数上描述了密码学，并认为它是一个群体行为。特别地，他研究了与RSA密码系统相关的组结构，并证明了存在一对不同的密钥，它们对消息进行相同的加密。在日本数学会举办的第20次民间讲座上，内田证明了许多周期结不具有delta解结数1。此外，他还推测，对于通常的解结操作，周期解结数不是1。上野研究了无限远处卡诺空间调和映射的狄利克雷问题。获得的结果在东京城市大学和山口大学举行的研究会议上宣读。

The purpose of our research is to investigate properties of the complex projective space from several points of view using various kinds of method. Each investigator has aimed to develop the research using his own special knowledge and have the following research results.Ogata, the head of this investigating team, has studied characterization and classification of surfaces with parallel mean curvature vector field in the complex projective space and proved that there exist minimal surfaces with constant Kaehler angle and non-constant Gaussian curvature in the 2-dimensional complex projective space.Hirabuki has proved Krull-Akizuki theorem for semigroups by using the method of proving theorem for rings arising from semigroup rings.It has constructed complex structures on the tangent bundle of the quaternion projective space by using the geometrical technique and proved that these complex structures correspond with the ones which were given by Furuya-Tanaka-Yoshizawa.Sawada has algebrically described cryptology and considered it as a group action. Specially he has examined the structure of groups related to RSA cryptosystems and showed the existence of a pair of different keys which encrypt messages equally. These results are read at the 20-th civil lecture promoted by Japan Mathematical Society,Uchida has proved that many periodical knots have not the DELTA-unknotting number 1. Moreover, he conjecture that the periodical unknotting number is not 1 for the usual unknotting operation.Ueno has studied the Dirichlet problems at infinity for harmonic maps between the Carnot spaces. The obtained results are read at the research meetings held at Tokyo Metropolitan Univ. and Yamaguchi Univ..