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**Biholomorphic mappings between bounded domains in \(C^ n\).**
*(English)*
Zbl 0867.32005

The main result of the paper is the following:

Theorem 1. Let \(f: \Omega\to D\) be a biholomorphic mapping between \(C^{2,\alpha}\) smooth bounded domains in \(\mathbb{C}^n\). Suppose that \(f\) is in \(C^{2,\alpha}(\overline{\Omega})\). Then \(f\) extends to a \(C^{2,\alpha}\)-diffeomorphism between \(\overline{\Omega}\) and \(\overline{D}\) \((\alpha>0)\).

This was proved by J. Fornaess [Pac. J. Math 74, 63-65 (1977; Zbl 0353.32026)] in the case when \(\alpha = 0\) and the domains are pseudoconvex. One of the key points in the proof is to show that a point \(p\in \partial\Omega\) is minimal (i.e. \(\partial \Omega\) does not contain a germ of a complex hypersurface through \(p\)) if and only if so is \(f(p)\in\partial D\).

Along with the main result, a version of the classical Hopf lemma is obtained for biholomorphic maps. A local version of Theorem 1 is also given, namely, it is shown that if \(f\in C^{2,\alpha}(\overline{\Omega})\) is an unbranched proper holomorphic map from \(\Omega\) to \(D\), then \(f\) extends to a local \(C^{2,\alpha}\) diffeomorphism between \(\overline \Omega\) and \(\overline{D}\). The method of the proof of Theorem 1 is used for proving that a proper holomorphic map \(f:\Omega\to D\) which is smooth up to the boundary does not vanish to infinite order at any minimal boundary point of \(\Omega\).

Theorem 1. Let \(f: \Omega\to D\) be a biholomorphic mapping between \(C^{2,\alpha}\) smooth bounded domains in \(\mathbb{C}^n\). Suppose that \(f\) is in \(C^{2,\alpha}(\overline{\Omega})\). Then \(f\) extends to a \(C^{2,\alpha}\)-diffeomorphism between \(\overline{\Omega}\) and \(\overline{D}\) \((\alpha>0)\).

This was proved by J. Fornaess [Pac. J. Math 74, 63-65 (1977; Zbl 0353.32026)] in the case when \(\alpha = 0\) and the domains are pseudoconvex. One of the key points in the proof is to show that a point \(p\in \partial\Omega\) is minimal (i.e. \(\partial \Omega\) does not contain a germ of a complex hypersurface through \(p\)) if and only if so is \(f(p)\in\partial D\).

Along with the main result, a version of the classical Hopf lemma is obtained for biholomorphic maps. A local version of Theorem 1 is also given, namely, it is shown that if \(f\in C^{2,\alpha}(\overline{\Omega})\) is an unbranched proper holomorphic map from \(\Omega\) to \(D\), then \(f\) extends to a local \(C^{2,\alpha}\) diffeomorphism between \(\overline \Omega\) and \(\overline{D}\). The method of the proof of Theorem 1 is used for proving that a proper holomorphic map \(f:\Omega\to D\) which is smooth up to the boundary does not vanish to infinite order at any minimal boundary point of \(\Omega\).

Reviewer: J.Davidov (Sofia)

### MSC:

32H02 | Holomorphic mappings, (holomorphic) embeddings and related questions in several complex variables |

32H35 | Proper holomorphic mappings, finiteness theorems |

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