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A new gap phenomenon for proper holomorphic mappings from B^n into B^N | | Xiaojun Huang
; Shanyu Ji
; Dekang Xu
; | | Date: |
2 May 2006 | | Journal: | Math. Res. Lett. 13 (2006). No 4, 509-523 | | Subject: | Complex Variables; Differential Geometry | | Abstract: | In this paper (Math. Res. Lett. 13 (2006). No 4, 509-523), the authors established a pseudo-normal form for proper holomoprhic mappings between balls in complex spaces with degenerate rank. This then was used to give a complete characterization for all proper holomorphic maps with geometric rank one, which, in particular, includes the following as an immediate application: Theorem: Any rational holomorphic map from B^n into B^N with $4le nle Nle 3n-4$ is equivalent to the D’Angelo map $$F_{ heta}(z’,w)=(z’,(cos heta)w,(sin heta)z_1w, ..., (sin heta)z_{n-1}w, (sin heta)w^2, 0’), 0le hetaleq pi/2.$$ It is a well-known (but also quite trivial) fact that any non-constant rational CR map from a piece of the sphere $partial {B^n}$ into the sphere $partial {B^N}$ can be extended as a proper rational holomoprhic map from $B^n$ into $B^N$ ($Nge nge 2$). By using the rationality theorem that the authors established in [HJX05], one sees that the the above theorem (and also the main theorem of the paper) holds in the same way for any non-constant $C^3$-smooth CR map from a piece of $partial {B^n}$ into $partial{B^N}$. The paper [Math. Res. Lett. 13 (2006). No 4, 509-523] was first electronically published by Mathematical Research Letters several months ago at its home website: this http URL (The pdf file of the printed journal version can also be downloaded at this http URL). | | Source: | arXiv, math/0605068 | | Services: | Forum | Review | PDF | Favorites | |
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