Mapping Bn into B2n-1
Mapping Bn into B2n-1
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DOI：
10.1007/s002220100140
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发表时间：
2001-08
影响因子：
3.1
通讯作者：
Xiaojun Huang;S. Ji
中科院分区：
文献类型：
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作者：
Xiaojun Huang;S. Ji
本文研究复空间中球间真全纯映射的分类问题。写出Bn={z∈ Cn：|z| < 1}和Prop（Bn，BN）是从Bn到BN的所有真全纯映射的集合.我们记得f，g∈ Prop（Bn，BN）称为等价的，如果存在元素σ∈ Aut（Bn）和τ∈ Aut（BN）使得f= τ <$g <$σ. Poincaré [Po]和亚历山大[Alx]的一个著名结果是：当N= n> 1时，任何f∈ Prop（Bn，Bn）都等价于恒等映射.对于N> n> 1的情况，由于发现了内部函数，很明显，解决Prop（Bn，BN）中的分类问题是不现实的。因此，人们关注映射的重要子类Rat（Bn，BN），即从Bn到BN的所有有理真全纯映射的集合。在这里，已经有许多非平凡和有趣的问题（[DA 1]）。第一个结果沿着这些路线是由于韦伯斯特[我们]，谁表明，大鼠（Bn，Bn+ 1）只有一个等价类n> 2。Faran [Fa 2]在较大的余维情况下证明了这也适用于Rat（Bn，BN）：N≤ 2n− 2。对于N≥ 2n− 1的情形，Rat（Bn，BN）的所有等价类的集合，记为R（n，N），具有来自[Fo 1]，[BER]的工作的真实的代数结构。然而，R（n，N）的具体描述仍然是相当神秘的一般。在[DA 2]中，D 'Angelo发现了从Bn到B2 n的一个连续的互不等价的多项式真嵌入族（见例3），这特别表明当N≥ 2n时，集合R（n，N）包含无穷多个元素。另一方面，在80年代初Faran [Fa 1]的一篇论文中，
In this paper, we are concerned with the classification problem of proper holomorphic maps between balls in complex spaces. Write Bn={z∈ Cn:| z|< 1} and Prop (Bn, BN) for the collection of all proper holomorphic maps from Bn into BN. We recall that f, g∈ Prop (Bn, BN) are said to be equivalent if there are elements σ∈ Aut (Bn) and τ∈ Aut (BN) such that f= τ◦ g◦ σ. It is a well-known result of Poincaré [Po] and Alexander [Alx] that when N= n> 1, then any f∈ Prop (Bn, Bn) is equivalent to the identity map. For the case of N> n> 1, due to the discovery of inner functions, it is clear that solving the classification problem in Prop (Bn, BN) is unrealistic. Therefore, one focuses on the important subclass of mappings, Rat (Bn, BN), the collection of all rational proper holomorphic mappings fromBn intoBN. And here, there are already many non-trivial and interesting questions ([DA1]).A first result along these lines is due to Webster [We], who showed that Rat (Bn, Bn+ 1) has only one equivalence class for n> 2. This was proven to hold also for Rat (Bn, BN) by Faran [Fa2] in the larger codimensional case: N≤ 2n− 2. For the case N≥ 2n− 1, the collection of all equivalence classes, denoted by R (n, N), of Rat (Bn, BN) carries a real algebraic structure from the work of [Fo1],[BER]. However, the specific description of R (n, N) remains to be quite mysterious in general. In [DA2], D’Angelo discovered a continuous family of mutually inequivalent polynomial proper embeddings from Bn into B2n (see Example 3), which in particular indicates that the set R (n, N) contains infinitely many elements when N≥ 2n. On the other hand, in a paper of Faran [Fa1] in the early 80’s, it was shown that