Families of Exposing Maps in Strictly Pseudoconvex Domains

Abstract

We prove that given a family (Gt) of strictly pseudoconvex domains varying in C 2 topology on domains, there exists a continuously varying family of exposing maps h t,ζ for all Gt at every ζ ∈ ∂Gt.

Splitting Lemma for Biholomorphic Mappings with Smooth Dependence on Parameters

“…We conjecture that the families α t,ζ , β t,ζ may be constructed to depend on a C l−1continuous way with respect to all variables. Remark 1.7 Some special case of Theorem 1.3 shall be one of the crucial devices in the construction of the C k -continuous family of exposing mappings of boundary points for the family of strictly pseudoconvex domains varying smoothly with the parameter, where the space of parameters is not necessarily compact (the paper in preparation); this would improve and extend the main result of [13], thus giving the comprehensive answer to the problem of the existence of such family, proposed in [1].…”

“…This is [4,Theorem 4.1] up to the type of the estimates (1.1), which are not necessarily linear in the quoted paper (the above formulation is [6, Theorem 9.7.1]). It is the crucial device in the construction of noncritical holomorphic functions and submersions from Stein manifolds ([4, Theorems 2.1 and 2.6]), or in the construction of the (families of the) exposing mappings of strictly pseudoconvex domains at boundary points (see [2,3,13], etc.). It is worth mentioning that the first application of this latter type was made in [5] in the problem of constructing proper holomorphic embeddings of Riemann surfaces to C 2 .…”

“…(2) We need the parametric solution of the additive Cousin problem with bounds and fulfilling some parametric interpolation condition (see Lemma 1.8). Concerning the solution without the interpolation condition, we have constructed one in [13]. However, it had the drawback that the solution is on slightly smaller domains than the input data.…”

“…3 must be linear isomorphisms. Had we used methods of [13], they would only become injections. This requires a parametric solution of the ∂-problem with extremely nice behaviour with respect to parameters.…”