Compactness of families of holomorphic mappings up to the boundary. (English) Zbl 0633.32020

Bell, S.

Complex analysis, Semin. University Park/Pa. 1986, Lect. Notes Math. 1268, 29-42 (1987).

[For the entire collection see Zbl 0614.00008.]
The paper deals with properties of limits, for the uniform convergence on compact subsets of \(\Omega_ 1\), of sequences of biholomorphic mappings \(f_ i: \Omega_ 1\to \Omega_ 2\), where \(\Omega_ 1\) and \(\Omega_ 2\) are bounded weakly pseudoconvex domains of finite type with \(C^{\infty}\) boundary. For instance, the following results are proved:
1) If \(f_ i: \Omega_ 1\to \Omega_ 2\) is a sequence of biholomorphic mappings which converge to \(f: \Omega\) \({}_ 1\to b\Omega_ 2\), then \((f_ i)\) converges uniformly to a constant map on compact subsets of \(\Omega_ 1\cup \Gamma\), where \(\Gamma\) is the set of strictly pseudoconvex boundary points of \(\Omega_ 1.\)
2) If, in addition, the sequence of inverse mappings \(F_ i=f_ i^{- 1}\) converges, there exist \(p_ 1\in b\Omega_ 1\) and \(p_ 2\in b\Omega_ 2\) such that \((f_ i)\) converges to the constant map \(f=p_ 2\), uniformly on compact subsets of \({\bar \Omega}_ 1-\{p_ 1\}\), and even \(C^{\infty}({\bar \Omega}_ 1-(S\cup \{p_ 1\})\), where S is the set of points z in \(b\Omega_ 1-\{p_ 1\}\) where \(K_ 1(z,p_ 1)=0\) \((K_ 1:\) Bergman kernel of \(\Omega_ 1).\)
The proofs use extension properties of the Bergman kernel up to the boundary.
Results for sequences of proper holomorphic mappings are also given, without proof.

Reviewer: G.Roos

MSC:

32T99	Pseudoconvex domains
32A25	Integral representations; canonical kernels (Szegő, Bergman, etc.)

Keywords:

properties of limits; sequences of biholomorphic mappings; weakly pseudoconvex domains; Bergman kernel; sequences of proper holomorphic mappings

Citations:

Zbl 0614.00008