A new gap phenomenon for proper holomorphic mappings from B^n into B^N

By Xiaoju Xu. 2006

88 citations recorded by the source corpus.

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 $4\le n\le N\le 3n-4$ is equivalent to the D'Angelo map $$F_{\theta}(z',w)=(z',(\cos\theta)w,(\sin\theta)z_1w, ..., (\sin\theta)z_{n-1}w, (\sin\theta)w^2, 0'), 0\le \theta\leq \pi/2.$$ 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 $B^N$ ($N\ge n\ge 2$). 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).

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/72031aee4066a62ed769c05d6fb20edaa21c9998

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In Applied Mathematics

  1. W. Yin
  2. S. Ji
  3. Xiaojun Huang
  4. Xiaoju Xu
  5. M. S. Baouendi
  6. P. Ebenfelt

In Geometry and Topology

  1. W. Yin
  2. S. Ji
  3. Xiaojun Huang
  4. Xiaoju Xu
  5. M. S. Baouendi
  6. P. Ebenfelt

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