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Characterization of Convex Domains with Noncompact Automorphism Group

机译:具有非紧同同构群的凸域的刻画

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摘要

The conformal mapping theorem of Riemann asserts that a simply connected domain in C, different from C, is biholomorphically equivalent to the open unit disc U = {z ∈C : |z| < 1}. Many authors have been interested in the generalization of this result in several complex variables (cf. [2; 3; 4; 9; 14]). The situation is quite different there: a small C~2 perturbation of the unit ball B~(n+1) in C~(n+1) can be nonequivalent to B~(n+1), even if it is simply connected. This shows that a domain in C~(n+1) is not completely described by its topological properties. Thus one must study the automorphism group of a domain to find a polynomial representation of it, that is, a rigid polynomial domain and a biholomorphic equivalence between our original domain and this rigid polynomial domain.
机译:Riemann的共形映射定理断言,与C不同,C中的简单连接域与开放单元圆盘U = {z∈C:| z | <1}。许多作者对将这个结果推广到几个复杂变量感兴趣(参见[2; 3; 4; 9; 14])。那里的情况大不相同:即使简单地将C〜(n + 1)中的单位球B〜(n + 1)的小C〜2扰动都等同于B〜(n + 1), 。这表明C〜(n + 1)中的一个域不能完全通过其拓扑特性来描述。因此,必须研究一个域的自同构群以找到它的多项式表示形式,即一个刚性多项式域以及我们原始域和该刚性多项式域之间的双全等值。

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