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 $(Σ, o)$ be the pointed Siegel hypersurface given by the defining equation $Re w - |z|^2 = 0$. If there is a holomorphic mapping $F:(M,o) \to (Σ,o)$ 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 $Aut (Σ,o)$.
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 $E_m = \{|z|^{2m} + |w|^2 < 1\}$.
Cartan-Thullen theorem is a basic one in the theory of analytic functions of several complex variables. It states that for any open set $U$ of ${\mathbb C}^k$, the following conditions are equivalent: (a) $U$ is a domain of existence, (b) $U$ is a domain of holomorphy and (c) $U$ is holomorphically convex. On the other hand, when $f \, (\, =(f_1,f_2,\cdots,f_n)\, )$ is a $\mathbb C^n$-valued function on an open set $U$ of $\mathbb C^{k_1}\times\mathbb C^{k_2}\times\cdots\times\mathbb C^{k_n}$, $f$ is said to be $\mathbb C^n$-analytic, if $f$ is complex analytic and for any $i$ and $j$, $i\not=…
We show that several spaces of holomorphic functions on a Riemann domain over a Banach space, including the nuclear and Hilbert-Schmidt bounded type, are locally $m$-convex Fréchet algebras. We prove that the spectrum of these algebras has a natural analytic structure, which we use to characterize the envelope of holomorphy. We also show a Cartan-Thullen type theorem.
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It is shown that any symplectic $2n\times 2n$-matrix, whose entries are complex holomorphic functions on a reduced Stein space, can be decomposed into a finite product of elementary symplectic matrices if and only if it is null-homotopic. Moreover, if this is the case, the number of factors can be bounded by a constant depending only on $n$ and the dimension of the space.
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