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Study of special domains

特殊领域研究

基本信息

批准号：
11640148
负责人：
SHIMIZU Satoru
金额：
$ 2.3万
依托单位：
Tohoku University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1999
资助国家：
日本
起止时间：
1999 至 2001
项目状态：
已结题

项目摘要

In this research, we have obtained the following results related to special domains.1. Concerning tube domains, we have established the fundamental result on the prolongation of complete polynomial vector fields on a tube domain, which is called the Prolongation Theorem. Besides, using the Prolongation Theorem, we have obtained a result on the characterization of n-dimensional abelian ideals in the Lie algebra of the affine automorphism group of a tube domain in the n-dimensional complex number space.2. Related to the study of Reinhardt domains or weakly pseudoconvex domains, we have made a study of the problem of characterizing generalized complex ellipsoids with spherical boundary points, and obtained the Riemann mapping theorem type result that most of such generalized complex ellipsoids coincide with the balls. Moreover, to clarify the aspect that the study of tube domains complements the study of Reinhardt domains, we have tried to give an another proof of this result by making use of the classification of spherical tube manifolds due to Dadok and Yang, and succeeded in the case of a class of generalized complex ellipsoids.3. Related to the study of the boundaries of special domains, we have investigated the CR structures. In particular, we have clarified a characteristic of set-theoretic representations of DR Lie algebras that correspond to the category dual to CR Lie algebras.4. As a study of torus actions, we have obtained the result that ample line bundles on nonsingular tone varieties are projectively normal. Also, we have made a detailed study of the two notions of stability for Fano manifolds - the K stability and the CM stability - introduced by Tian related to the Hitchin-Kobayashi correspondence for manifolds.5. Related to the geometry of the boundaries of special domains, we have given various generalizations of constant mean curvature surfaces.

在本研究中，我们取得了以下与特殊领域相关的成果： 1．关于管域，我们建立了管域上完全多项式向量场延长的基本结果，称为延长定理。此外，利用延长定理，我们得到了n维复数空间管域仿射自同构群李代数中n维阿贝尔理想的刻画结果。 2．与Reinhardt域或弱赝凸域的研究相关，我们对用球面边界点表征广义复椭球体的问题进行了研究，得到了大多数此类广义复椭球体与球体重合的黎曼映射定理式结果。此外，为了阐明管域研究与Reinhardt域研究的补充，我们尝试利用Dadok和Yang的球面管流形分类对这一结果进行了另一种证明，并在一类广义复杂椭球体的情况下取得了成功。 3．与特殊域边界的研究相关，我们研究了CR结构。特别地，我们阐明了DR李代数的集合论表示的特征，其对应于CR李代数的对偶范畴。4．作为对环面作用的研究，我们得到的结果是，非奇异音调品种上的大量线束在投影上是正态的。此外，我们还详细研究了Tian引入的与流形的Hitchin-Kobayashi对应相关的Fano流形稳定性的两个概念——K稳定性和CM稳定性。5．与特殊域边界的几何形状相关，我们给出了常平均曲率曲面的各种推广。