Splitting Lemma for Biholomorphic Mappings with Smooth Dependence on Parameters
- PDF / 348,415 Bytes
- 16 Pages / 439.37 x 666.142 pts Page_size
- 25 Downloads / 174 Views
Splitting Lemma for Biholomorphic Mappings with Smooth Dependence on Parameters Arkadiusz Lewandowski1 Received: 21 April 2020 © The Author(s) 2020
Abstract l−1 We prove that the mappings obtained in Forstneriˇc splitting lemma vary in a C 2 continuous way if only the input family of biholomorphic mappings close to Id (and their domains) is C l -continuous (see Theorem 1.3 for a precise formulation). Keywords Strictly pseudoconvex domains · Forstneriˇc splitting lemma · Cousin problems Mathematics Subject Classification Primary 32H02 · Secondary 32T15
1 Introduction Let X be a complex manifold and let dist be a Riemannian distance function on X . For a set U ⊂ X and a mapping γ : U → X put distU (γ , Id) := sup dist(γ (x), x), x∈U
where Id: X → X is an identity mapping. The so-called splitting lemma for biholomorphic maps close to identity due to Forstneriˇc reads as follows: Theorem 1.1 Let A, B ⊂ X be compact sets such that A ∪ B is a Stein compact and A \ B ∩ B \ A = ∅. For an open set C containing A ∩ B, there exist open sets A , B , C such that A ⊂ A , B ⊂ B , A ∩ B ⊂ C ⊂ A ∩ B ⊂ C, and constants ε0 > 0, c > 0, such that the following holds true: for any injective holomorphic
The author was supported by the Grant UMO-2017/26/D/ST1/00126 financed by the National Science Centre, Poland.
B 1
Arkadiusz Lewandowski [email protected] Institute of Mathematics, Faculty of Mathematics and Computer Science, Jagiellonian University, Łojasiewicza 6, 30-348 Kraków, Poland
123
A. Lewandowski
mapping γ : C → X with distC (γ , Id) =: ε < ε0 there exist injective holomorphic mappings α : A → X , β : B → X , depending continuously on γ , with dist A (α, Id) < cε, dist B (β, Id) < cε,
(1.1)
and such that γ ◦ α = β on C . If, moreover, A ∩ B is a Stein compact and X ⊂ X is a closed complex subvariety such that X ∩ C = ∅, then α and β can be chosen tangent to the identity to any given finite order along X . 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 C2 . From now on we restrict ourselves to the case where X = Cn and the distance is the Euclidean one, and to the situation where the interior of A ∪ B is a bounded strictly pseudoconvex domain. The latter three papers revealed the need for versions of splitting lemma with (smooth) parameters in the situation of this kind. Namely, it is desirable to know that if the mappings γt and their domains of definition vary with certain regularity with parameter t, then the output mappings αt an
Data Loading...
