Local convexifiability of some rigid domains. (English) Zbl 1108.32017

Slovák, Jan (ed.) et al., The proceedings of the 24th winter school “Geometry and physics”, Srní, Czech Republic, January 17–24, 2004. Palermo: Circolo Matemático di Palermo. Supplemento ai Rendiconti del Circolo Matemático di Palermo. Serie II 75, 251-257 (2005).
On some pseudoconvex domains with smooth boundary in \(\mathbb{C}^n\) there do not exist holomorphic reproducing support functions in the sense of Cauchy-Leray integral representations. It is therefore worthwile to characterise those domains which admit such functions. A trivial case is when the domain is convex. But it is still an open problem to characterise the domains which are locally biholomorphically equivalent to convex ones. This problem is much easier if \(n= 2\). Since it is known that then domains of finite type \(\leq 4\) admit local holomorphic support functions the author proposes himself to study convexibility of this case. His main result is an affirmative answer with the additional hypothesis that the domain is rigid and has real analytic boundary. However, it should be stressed that it is not a trivial problem to know if a given domain is rigid up to a local transformation.
The proof starts by transforming the defining function of the domain in a normal form. Since the author already has treated the above problem in the generic case with respect to this normal form he only has to deal with a special case where the normal form is quite explicit, at least in the main terms. Then the author decomposes the normal form in a sum of simpler expressions which are then convexified by a sequence of sophisticated local holomorphic transformations.
For the entire collection see [Zbl 1074.53001].


32T25 Finite-type domains
32F32 Analytical consequences of geometric convexity (vanishing theorems, etc.)
32A26 Integral representations, constructed kernels (e.g., Cauchy, Fantappiè-type kernels)