  In this article, we consider a bounded pseudoconvex domain in ${\bf C}^2$
satifying:
  (a) it admits a proper holomorphic mapping $f$ onto the unit ball $B^2$, and
  (b) it is simply connected and has a real analytic boundary.
  According to [Barletta-Bedford, Indiana U. Math. J, 39(1985), 315-338], the
strong pseudconvexity of $B^2$ alone yields that such a domain is "weakly
spherical" at the boundary points that are at the same time a smooth point of
the branch locus $Z_{df} = \{\det(J_{\bf C} f) = 0\}$. (Notice that
[Diederich-Fornaess, Math. Ann., 282 (1988), 681-700] implies that $f$ as well
as $Z_{df}$ extends holomorphically across the boundaries.)
  Our main contribution in this paper is that we have discovered a stronger
rigidity (both local and global) in case the target domain is the unit ball.
The main results are:
  THEOREM ("Local Rigidity"): Let $(M,o)$ be a real analytic normalized weakly
spherical pointed CR hypersurface in ${\bf C}^2$ of order $k_0 > 1$. Let
$(\Sigma, o)$ be the pointed Siegel hypersurface given by the defining equation
$Re w - |z|^2 = 0$. If there is a holomorphic mapping $F:(M,o) \to (\Sigma,o)$
for which $o$ is a regular branch point, then
  (1) $(M,o)$ is defined by the equation $Re w - |z|^{2k_0} = 0$, and
  (2) $F(z,w)$ is equivalent to $(z,w) \mapsto (z^{k_0},w)$ up to a composition
with elements in $Aut (M,o)$ and $Aut (\Sigma,o)$.
  THEOREM ("Global Rigidity"): Let $D$ and $f:D \to B^2$ be as above, and let
$f$ be generically $m$-to-1. Assume that its branch locus $Z_{df}$ admits an
analytic component $V$ with the following properties:
  (1) $f$ is locally a $m$-to-1 branched covering with branch locus $V$ at
every point of $V \cap \partial D$;
  (2) $V \cap \partial D$ is connected and contains no singular point of the
variety $Z_{df}$.
  Then $D$ is biholomorphic to $E_m = \{|z|^{2m} + |w|^2 < 1\}$.
