×

Proper holomorphic maps from weakly pseudoconvex domains. (English) Zbl 0716.32017

Let D be a bounded pseudoconvex domain in \({\mathbb{C}}^ 2\) with real- analytic boundary. The authors prove the existence of a proper holomorphic mapping from D into the unit polydisc in \({\mathbb{C}}^ 3\). They also prove that there exists a uniformly continuous proper holomorphic mapping from D into the unit ball in \({\mathbb{C}}^ 3\). The techniques employed in the construction are related to those used in proofs of existence of inner functions.
Reviewer: M.Klimek

MSC:

32H35 Proper holomorphic mappings, finiteness theorems
32T99 Pseudoconvex domains
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] A. B. Aleksandrov, The existence of inner functions in a ball , Mat. Sb. (N.S.) 118(160) (1982), no. 2, 147-163, 287, English transl. in Math. USSR-Sb. 46 (1983), 143-159. · Zbl 0503.32001
[2] A. B. Aleksandrov, Proper holomorphic mappings from the ball to the polydisk , Dokl. Akad. Nauk SSSR 286 (1986), no. 1, 11-15, English transl. in Soviet Math. Dokl. 33 (1986), 1-5. · Zbl 0602.32008
[3] E. Bedford and J. E. Fornaess, A construction of peak functions on weakly pseudoconvex domains , Ann. of Math. (2) 107 (1978), no. 3, 555-568. JSTOR: · Zbl 0392.32004
[4] T. Bloom, \(\mathcal C^\infty \) peak functions for pseudoconvex domains of strict type , Duke Math. J. 45 (1978), no. 1, 133-147. · Zbl 0376.32014
[5] T. Bloom and I. Graham, A geometric characterization of points of type \(m\) on real submanifolds of \(\mathbf C\spn\) , J. Differential Geometry 12 (1977), no. 2, 171-182. · Zbl 0436.32013
[6] K. Diederich and J. E. Fornaess, Pseudoconvex domains with real-analytic boundary , Ann. Math. (2) 107 (1978), no. 2, 371-384. · Zbl 0378.32014
[7] A. Dor dissertation, Princeton University, 1988.
[8] J. E. Fornaess, Peak points on weakly pseudoconvex domains , Math. Ann. 227 (1977), no. 2, 173-175. · Zbl 0346.32026
[9] J. E. Fornaess and N. Øvrelid, Finitely generated ideals in \(A(\Omega )\) , Ann. Inst. Fourier (Grenoble) 33 (1983), no. 2, 77-85. · Zbl 0489.32013
[10] F. Forstnerič, Embedding strictly pseudoconvex domains into balls , Trans. Amer. Math. Soc. 295 (1986), no. 1, 347-368. JSTOR: · Zbl 0594.32024
[11] M. Hakim, Applications holomorphes propres continues de domaines strictement pseudoconvexes de \(\mathbf C^ n\) dans la boule unité de \(\mathbf C^ n+1\) , Duke Math. J. 60 (1990), no. 1, 115-133. · Zbl 0694.32010
[12] M. Hakim and N. Sibony, Fonctions holomorphes bornées sur la boule unité de \(\mathbf C\spn\) , Invent. Math. 67 (1982), no. 2, 213-222. · Zbl 0475.32007
[13] G. M. Henkin and J. Leiterer, Theory of Functions on Complex Manifolds , Monographs in Mathematics, vol. 79, Birkhauser Verlag, Basel, 1984. · Zbl 0573.32001
[14] J. J. Kohn and L. Nirenberg, A pseudo-convex domain not admitting a holomorphic support function , Math. Ann. 201 (1973), 265-268. · Zbl 0248.32013
[15] S. Lojasiewicz, Triangulation of semi-analytic sets , Ann. Scuola Norm. Sup. Pisa (3) 18 (1964), 449-474. · Zbl 0128.17101
[16] E. Løw, A construction of inner functions on the unit ball in \(\mathbf C\spp\) , Invent. Math. 67 (1982), no. 2, 223-229. · Zbl 0528.32006
[17] E. Løw, Inner functions and boundary values in \(H^\infty (\Omega )\) and \(A(\Omega )\) in smoothly bounded pseudoconvex domains , Math. Z. 185 (1984), no. 2, 191-210. · Zbl 0526.32017
[18] E. Løw, The ball in \(\mathbbC^n\) is a closed complex submanifold of a polydisc , Invent. Math. 83 (1986), no. 3, 405-410. · Zbl 0617.32039
[19] E. Løw, Embeddings and proper holomorphic maps of strictly pseudoconvex domains into polydiscs and balls , Math. Z. 190 (1985), no. 3, 401-410. · Zbl 0584.32048
[20] A. Noell, Interpolation from curves in pseudoconvex boundaries , Michigan Math. J. (to appear). · Zbl 0719.32013
[21] N. Sibony, Sur le plongement des domaines faiblement pseudoconvexes dans des domaines convexes , Math. Ann. 273 (1986), no. 2, 209-214. · Zbl 0573.32023
[22] B. Stensønes, Proper holomorphic mappings from strongly pseudoconvex domains in \(\mathbbC^2\) to the unit polydisc in \(\mathbbC^3\) , Math. Scand. (to appear). · Zbl 0706.32010
[23] B. Stensønes, Constructions of proper holomorphic mappings from strongly pseudoconvex domains in \(\mathbbC^n\) into polydiscs , · Zbl 0706.32010
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.