The exhaustion method for skew cylinders (Q2761460)

This paper treats the question of simultaneous uniformization of holomorphic families \(M=\{M_p:p\in B\}\) of simply connected Riemann surfaces \(M_p\). Taking \(M\) to be a Euclidean ball in \({\mathbb C}^2\) and comparing it with the unit polydisc \(U\times U\), we see that it is not to be expected that such families \(M\) are holomorphically trivial, i.e., \(M\) may not be bundle biholomorphic (even just biholomorphic) to \(B\times M_{p_0}\), where \(M_{p_0}\) is the fiber type of \(M\). But we may ask for just a fibered holomorphic embedding of \(M\) into the simplest family \(B\times{\mathbb CP}^1\) or \(B\times{\mathbb C}\). In the first case we call our family \(M\) uniformizable, in the second strongly uniformizable. NEWLINENEWLINENEWLINEThe following question seems well-posed: Is every Stein skew cylinder uniformizable? How about if in addition \(M\) has a smooth strictly pseudoconvex boundary? NEWLINENEWLINENEWLINEHere a skew cylinder \(M\) is a special holomorphic family defined as follows: \(M\) is (the total space of) a holomorphic fiber bundle over a complex manifold \(B\), which has a simply connected Riemann surface \(E\) for fiber type, and \(M\) admits a holomorphic section \(s:B\to M\). A good source for skew cylinders is taking leafwise universal covers of foliations by Riemann surfaces (e.g., in holomorphic ordinary differential equations). NEWLINENEWLINENEWLINEThe above question of uniformizability of skew cylinders is open in general even in the smooth strictly pseudoconvex case (which case exhibits certain stability under small deformations), but as it is shown in \S 3 at least one can reduce by exhaustion the general case of uniformizability to the smooth strictly pseudoconvex case. The main contribution the paper is the following Exhaustion Theorem, whose proof occupies about 17 pages. NEWLINENEWLINENEWLINEExhaustion Theorem. Let the base \(B\) of our skew cylinder \(M\) be a contractible pseudoconvex domain in some \({\mathbb C}^n\), then if \(M\) is a Stein manifold, then \(M\) contains an exhausting sequence of skew cylinders \(M_i\) which have smooth pseudoconvex boundaries transverse to the fibers of \(M\). NEWLINENEWLINENEWLINEThe proof of the Exhaustion Theorem is based on a very long and involved geometric, analytic procedure, which draws upon a detailed analysis of singularities in stratifications of semianalytic and subanalytic sets that come into play. (One needs to chew through a lot of kasha before one gets to the raisins.) NEWLINENEWLINENEWLINEThe paper concludes with the above mentioned reduction and proves the following. NEWLINENEWLINENEWLINEProposition 9. If a skew cylinder \(M\) admits an exhausting sequence of strongly uniformizable skew cylinders \(M_i\), then \(M\) itself is strongly uniformizable. NEWLINENEWLINENEWLINEProposition 10. Suppose that a skew cylinder \(M\) contains an exhausting sequence of uniformizable skew cylinders \(M_i\). Then \(M\) is also uniformizable. NEWLINENEWLINENEWLINE(The Exhaustion Theorem provides us with examples to which the above propositions apply.) The Russian and its English translation by the author are identical except that in the English version some of the references are anglicized. Reference [2] has been published in the meanwhile: \textit{Yu. S. Ilyashenko} [Topol. Methods Nonlinear Anal. 11, No. 2, 361-373 (1998; Zbl 0927.32020)]. Reference [9] by \textit{Z. Denkowska} and \textit{J. Stasica} [`Ensembles sous-analytiques...', (1991)] was not yet published. NEWLINENEWLINENEWLINEThe paper under review (despite its very technical nature at places) is very readable and was written in the best tradition of the Russian school.