Proper holomorphic mappings between real-analytic pseudoconvex domains in \({\mathbb{C}}^ n\). (English) Zbl 0661.32025

The authors provide a “simple proof” that a proper holomorphic mapping between pseudoconvex domains D and D’ of \({\mathbb{C}}^ n\) with real- analytic boundaries extends to a proper holomorphic map between neighborhoods of the closures of D and D’. Although the authors describe the proof as “simple”, their arguments are technically rather involved. Heavy use is made of the implicit function theorem and formal power series techniques, as well as, the idea of “complexification of a real hypersurface” as developed by S. Webster, to whom the authors give special credit.
Reviewer: G.Harris


32V40 Real submanifolds in complex manifolds
32H35 Proper holomorphic mappings, finiteness theorems
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[1] D’Angelo, J.: Real hypersurfaces, orders of contact, and applications. Ann. Math.115, 615-637 (1982) · Zbl 0488.32008
[2] Baouendi, M.S., Bell, S., Rothschild, L.P.:CR mappings of finite multiplicity and extension of proper holomorphic mappings. Bull. AMS16, 265-270 (1987) · Zbl 0627.32016
[3] Baouendi, M.S., Bell, S.R., Rothschild, L.P.: Mappings of three-dimensionalCR-manifolds and their holomorphic extension. Duke Math. J.56, 503-530 (1988) · Zbl 0655.32015
[4] Baouendi, M.S., Jacobowitz, H., Treves, F.: On the analyticity ofCR mappings. Ann. Math.122, 365-400 (1985) · Zbl 0583.32021
[5] Bedford, E., Bell, S.: Extension of proper holomorphic mappings past the boundary. Manuscr. Math.50, 1-10 (1985) · Zbl 0583.32044
[6] Bedford, E., Fornæss, J.E.: Extension ofCR-functions from weakly pseudoconvex boundaries. Mich. Math. J.25, 259-262 (1978) · Zbl 0401.32007
[7] Bell, S.: Proper holomorphic mappings and the Bergman projection. Duke Math. J.48, 167-175 (1981) · Zbl 0465.32014
[8] Bell, S.: Analytic hypoellipticity of the \(\bar \partial\) -Neumann problem and extendability of holomorphic mappings. Acta Math.147, 109-116 (1981) · Zbl 0475.32010
[9] Bell, S., Catlin, D.: Boundary regularity of proper holomorphic mappings. Duke Math. J.49, 385-396 (1982) · Zbl 0486.32014
[10] Catlin, D.: Necessary conditions for subellipticity of the \(\bar \partial\) -Neumann problem. Ann. Math.117, 147-171 (1983) · Zbl 0552.32017
[11] Derridj, M.: Prolongement local d’applications holomorphes propres en des points faiblement pseudoconvexes. Preprint Orsay 1984
[12] Derridj, M.: Le preincipe de réflexion en des points de faible pseudo-convexité pour des applications holomorphes propres. Invent. Math.79, 197-215 (1985) · Zbl 0554.32011
[13] Derridj, M.: Multitype et prolongement d’applications holomorphes. Preprint Orsay 1985
[14] Derridj, M.: Sur le prolongement d’applications holomorphes. Sémin. Goualouic-Schwartz, Equations Dériv. Partielles, exposé XVI (1985-86)
[15] Diederich, K., Fornæss, J.E.: Pseudoconvex domains: Existence of Stein neighborhoods. Duke Math. J.44, 642-662 (1977) · Zbl 0381.32014
[16] Diederich, K., Fornæss, J.E.: Pseudoconvex domains with real-analytic boundary. Ann. Math.107, 371-384 (1978) · Zbl 0378.32014
[17] Diederich, K., Fornæss, J.E.: Boundary regularity of proper holomorphic mappings. Invent. Math.67, 363-384 (1982) · Zbl 0501.32010
[18] Diederich, K., Webster, S.M.: A reflection principle for degenerate real hypersurface. Duke Math. J.47, 835-843 (1980) · Zbl 0451.32008
[19] Han, C.K.: Analyticity ofCR-equivalences between some real hypersurfaces inC n with degenerate Levi forms. Invent. Math.73, 51-69 (1983) · Zbl 0517.32007
[20] Kohn, J.J.: Subellipticity of the \(\bar \partial\) -Neumann problem on pseudoconvex domains: sufficient conditions. Acta Math.142, 79-122 (1979) · Zbl 0395.35069
[21] Lewy, H.: On the boundary behavior of holomorphic mappings. Acad. Naz. Linc.35, 1-8 (1977) · Zbl 0377.31008
[22] Maingot, S.: Prolongement d’applications holomorphes propres. C.R. Acad. Sci. Paris303, 399-402 (1986) · Zbl 0597.32015
[23] Pincuk, S.J.: On the analytic continuation of biholomorphic mappings. Mat. Sb.98, 416-435 (1975); Math. USSR Sb.27, 375-392 (1975)
[24] Trépreau, J.M.: Sur le prolongement des fonctionsCR definis sur une hypersurface réele de classeC 2 dansC n . Invent. Math.83, 583-592 (1986) · Zbl 0586.32016
[25] Webster, S.M.: On the mapping problem for algebraic real hypersurfaces. Invent. Math.43, 53-68 (1977) · Zbl 0355.32026
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