A new gap phenomenon for proper holomorphic mappings from B^n into B^N
  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: http://www.mrlonline.org/mrl/0000-000-00/Huang-Ji-Xu2.pdf. (The pdf file of the printed journal version can also be downloaded at http://www.math.uh.edu/~shanyuji/rank1.pdf). 
Xiaojun Huang (Rutgers University) Shanyu Ji (University of Houston)   Dekang Xu (University of Houston)