Critical sets of proper holomorphic mappings
  It is shown that if a proper holomorphic map $f: \mathbb C^n \to \mathbb C^N$, $1<n\le N$, sends a pseudoconvex real analytic hypersurface of finite type into another such hypersurface, then any $n-1$ dimensional component of the critical locus of $f$ intersects both sides of $M$. We apply this result to the problem of boundary regularity of proper holomorphic mappings between bounded domains in $\mathbb C^n$. 
Sergey Pinchuk and Rasul Shafikov
math.CV