## 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:

  E. Bedford and S. Bell,Extension of proper holomorphic mappings past the boundary, Manuscr. Math.50 (1985), 1–10 · Zbl 0583.32044  M.S. Baouendi and L. P. Rothschild,A generalized complex Hopf lemma and its applications to CR mappings, Invent. Math111 (1993), 331–348 · Zbl 0781.32021  M.S. Baouendi and L. P. Rothschild,Normal forms for generic manifolds and holomorphic extension of CR functions, J. Differ. Geom.25 (1987), 431–467 · Zbl 0629.32016  K. Diederich and J. Fornaess,Pseudoconvex domains, existence of Stein neighborhood, Duke. Math. J.44 (1977), 641–662 · Zbl 0381.32014  K. Diederich, J. Fornaess and Z. Ye,Biholomorphisms in dimension two, J. Geometric Analysis (1995)  K. Diederich and S. Pincuk,The inverse of a CR-homeomorphism is CR, Inter. J. Math.4 (1993), 379–394 · Zbl 0784.32021  J. Fornaess,Biholomorphic mappings between weakly pseudoconvex domains, Pac. J. Math74 (1977), 63–65 · Zbl 0353.32026  X. Huang and S. Krantz,A unique continuation problem for holomoprhic mappings. Comm. in P. D. E.18 (1993), 241–263 · Zbl 0781.32018  X. Huang, S. Krantz, D. Ma, Y. Pan,A Hopf Lemma for holomorphic functions, Complex Variables26 (1995), 273–276 · Zbl 0837.30019  X. Huang and Y. Pan,On proper holomorphic mappings between real analytic domains in C n , Duke Math. Journal, to appear  Y. Pan,Real analyticity of CR homeomorphsims between real analytic hypersurfaces in C 2. Proc. A. M. S.123 (1995), 373–380 · Zbl 0813.32017  Y. Pan,A characterization of the finite multiplicity of a CR map, Michigan Journal of Math. (to appear) · Zbl 0873.32014  S. Pincuk,CR transformations of real manifolds in C n , Indiana. Univ. Math. J.41 (1992), 1–16 · Zbl 0766.32021  A. Tumanov,Extending CR functions on manifolds of finite type to a wedge, Mat. Sbornik136 (1988), 128–139  J. Trepreau,Sur le prolongement holomorphe des functions CR defins sur une hypersurfsce reelle de classe C 2 dans C n , Invent. Math.83, (1986), 583–592 · Zbl 0586.32016  W. Rudin,Function Theory in the unit ball of C n , Springer-Verlag · Zbl 0495.32001  H. Whitney,Complex Analytic Varieties, Addison-Wesley · Zbl 0265.32008
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