## 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 $$\Omega$$.
Reviewer: J.Davidov (Sofia)

### MSC:

 32H02 Holomorphic mappings, (holomorphic) embeddings and related questions in several complex variables 32H35 Proper holomorphic mappings, finiteness theorems

### Citations:

Zbl 0371.32015; Zbl 0353.32026
Full Text:

### References:

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