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Rigidity of holomorphic mappings and a new Schwarz lemma at the boundary. (English) Zbl 0807.32008

Vorliegende Arbeit studiert eine “Randversion” des Schwarz’schen Lemmas. Für jede holomorphe Abbildung \(f:B_ n \to B_ n\) \((B_ n\) bezeichne die Einheitskugel im \(\mathbb{C}^ n)\), die nahe eines Randpunktes von vierter Ordnung mit der Identität von \(B_ n\) übereinstimmt, gilt: \(f \equiv \text{id}_{B_ n}\). Unter Benutzung des Einbettungssatzes von Fornaess und Ergebnissen von Lempert wird obige Aussage auch für streng pseudokonvexe Gebiete bewiesen.
Bekannt ist [vgl. H. Alexander, Math. Ann. 209, 249-256 (1974; Zbl 0281.32019)], daß jede biholomorphe Abbildung \(\varphi:U \cap B_ n \to U' \cap B_ n (n \geq 2)\), die sich zu einer \(C^ 2\)-Abbildung auf den Rand fortsetzen läßt, Restriktion eines Automorphismus von \(B_ n\) ist. Obige Resultate werden benutzt, um Alexander’s Resultate wie folgt zu verschärfen: Sei \(\varphi : B_ n \to B_ n (n \geq 2)\) holomorph, und sei vorausgesetzt, daß \(\varphi\) sich nahe \(P \in \partial B_ n\) und \(\varphi (P) \in \partial B_ n\) als \(C^ 6\)- Abbildung auf den Rand fortsetzen läßt. Schmiegt sich dann \(\varphi (\partial B_ n)\) in \(\varphi (P)\) “hinreichend stark” an \(\partial B_ n\), so ist \(\varphi\) nahe \(P\) “fast” eine biholomorphe Abbildung von \(B_ n\). Genauer: \(\varphi (z) \equiv \psi (z) + 0 (| z - P |^ 3)\) mit \(\psi \in \operatorname{Aut} B_ n\).
Reviewer: P.Pflug (Vechta)

MSC:

32A40 Boundary behavior of holomorphic functions of several complex variables
32T99 Pseudoconvex domains
32H02 Holomorphic mappings, (holomorphic) embeddings and related questions in several complex variables

Citations:

Zbl 0281.32019
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References:

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