On a linearity problem for proper holomorphic maps between balls in complex spaces of different dimensions
On a linearity problem for proper holomorphic maps between balls in complex spaces of different dimensions
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DOI：
10.4310/jdg/1214425024
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发表时间：
1999
影响因子：
2.5
通讯作者：
Xiaojun Huang
中科院分区：
文献类型：
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作者：
Xiaojun Huang
在多复变的一个重要发展中，庞加莱[26]发现C2中单位球的两个开片之间的任何双全纯映射都是C2中单位双球B2的某个自同构的限制。这一现象在单复变情形下明显地不成立，而揭示了多元全纯映射的强刚性性质。后来，Tanaka等人（见[8]，[28]）将这一结果推广到任何维数的情况。亚历山大在他的著名论文[1]，[2]中进一步证明了Cn（n > 1）中球的任何真全纯自映射都是自同构，从而完成了对同一复空间中球之间的真全纯映射的理解的一系列研究。1978年，韦伯斯特[31]利用Cartan-Chern-Moser [8]理论，再次研究了从单位n-球Bn = {z ∈ Cn：|z|二、在这里，我们回想一下，从Bn到BN的真全纯映射称为全测地嵌入（或线性嵌入），如果存在自同构σ ∈ Aut（Bn）和τ ∈ Aut（BN）使得τ ∈ f <$σ =（id，0）。在随后的论文中，
In an important development of several complex variables, Poincaré [26] discovered that any biholomorphic map between two open pieces of the unit sphere in C2 is the restriction of a certain automorphism of B2, the unit two-ball in C2. This phenomenon fails obviously in one complex variable and reveals a strong rigidity property of holomorphic mappings in several variables. Later, Tanaka, etc (see [8], [28]) extended this result to any dimensional case. Alexander, in his famous papers [1], [2], further proved that any proper holomorphic selfmapping of the ball in Cn (n > 1) is an automorphism, thus finishing off a line of research towards the understanding of proper holomorphic mappings between balls in the same complex space. In 1978, using the Cartan-Chern-Moser [8] theory, Webster [31] took up again the problem of considering a proper holomorphic mapping f from the unit n-ball Bn = {z ∈ Cn : |z| 2. Here, we recall that a proper holomorphic map from Bn into BN is called a totally geodesic embedding (or a linear embedding) if there exist automorphisms σ ∈ Aut(Bn) and τ ∈ Aut(BN ) such that τ ◦ f ◦ σ = (id, 0). In a subsequent paper,