Factorization of proper holomorphic mappings through Thullen Domains
Abstract
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 [BarlettaBedford, Indiana U. Math. J, 39(1985), 315338], 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 [DiederichFornaess, Math. Ann., 282 (1988), 681700] 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 $m$to1. Assume that its branch locus $Z_{df}$ admits an analytic component $V$ with the following properties: (1) $f$ is locally a $m$to1 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 $E_m = \{z^{2m} + w^2 < 1\}$.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 April 1997
 DOI:
 10.48550/arXiv.math/9704201
 arXiv:
 arXiv:math/9704201
 Bibcode:
 1997math......4201K
 Keywords:

 Mathematics  Complex Variables;
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