A study on the structure of complex ellipsoids

本研究的主要目的是解决以下基本猜想：“设D是n维复欧氏空间C^n中的有界域，具有非紧自同构群Aut（D）.则D必双全纯于某个复椭球域E”。针对本研究，我们进行了以下研究：1.儿玉研究了给定复椭球E的所有非唯一点组成的集合的结构，并将他的思想应用于具有球形边界点的复椭球的特征化. Fujimoto研究了复射影空间P^nc中全纯映射的Nevanlinna理论，得到了P^3c中8.3次双曲超曲面的一些新结果。Shimizu研究了C^n中管域T_Ω上多项式向量场的李代数。他获得了延长定理等向量场和解决了等价问题管域。

The main purpose of this research is to solve the following fundamental con jecture : "Let D be a bounded domain in an n-dimensional complex Euclidean space C^n with non-compact automorphism group Aut(D). Then D is necessarily biholomorphic to some complex ellipsoidal domain E". Concerning this research, we studied and obtained the following :1. Kodama studied the structure of the set consisting of all non-unbilical points of a given complex ellipsoid E, and he applied his ideas to the characteri zation of complex ellipsoids with spherical boundary points.2. Fujimoto studied Nevanlinna theory of holomorphic mappings into the complex projective spaces P^nc, and obtained some new results on hyperbolic hypersurfaces in P^3c of degree 8.3. Shimizu studied the Lie algebra of polynomial vector fields on a tube domain T_Ω in C^n. He obtained Prolongation Theorem for such vector fields and solved the equivalence problem for tube domains.