Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Continuous proper holomorphic maps into bounded domains

Author: Avner Dor
Journal: Proc. Amer. Math. Soc. 119 (1993), 1145-1155
MSC: Primary 32H35; Secondary 32H02
MathSciNet review: 1162088
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Abstract: A continuous proper holomorphic map is constructed from the unit ball of $ {\mathbb{C}^N}$ to a smooth bounded domain in $ {\mathbb{C}^M}(2 \leqslant N \leqslant M - 1)$. The construction is done at a strongly convex corner of the target domain. At each stage the map is pushed farther into the boundary in a direction almost tangent to the boundary at a close vicinity. The close point property is employed, along with suitable peak functions, to obtain a minimal codimension. It appears to be close to the most general construction that can be made by summation of peak functions.

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  • [D1] A. Dor, Proper holomorphic maps between balls in one co-dimension, Ark. Mat. 28 (1990), 49-100. MR 1049642 (91g:32033)
  • [D2] -, Lifting of proper holomorphic maps, preprint.
  • [F] F. Forstnerič, Embedding strictly pseudoconvex domains into balls, Trans. Amer. Math. Soc. 295 (1986), 347-368. MR 831203 (87k:32052)
  • [FG] F. Forstnerič and J. Globevnik, Discs in pseudoconvex domains, preprint, 1991. MR 1144617 (92j:32100)
  • [G] J. Globevnik, Relative embeddings of discs into convex domains, Invent. Math. 98 (1989), 331-350. MR 1016268 (90j:32029)
  • [H] M. Hakim, Applications propres continues de domaines strictement pseudoconvexes de $ {\mathbb{C}^n}$ dans la boule unit $ {\mathbb{C}^{n + 1}}$, Duke Math J. 60 (1990), 115-133. MR 1047118 (91d:32035)
  • [HS] M. Hakim and N. Sibony, Fonction holomorphes sur la boule unite de $ {\mathbb{C}^n}$, Invent. Math. 67 (1982), 213-222. MR 665153 (84j:32008a)
  • [L1] E. Løw, A construction of inner functions on the unit ball in $ {\mathbb{C}^p}$, Invent. Math. 67 (1982), 223-229. MR 665154 (84j:32008b)
  • [L2] -, Embeddings and proper holomorphic maps of strictly pseudoconvex domains into polydiscs and balls, Math. Z. 190 (1985), 401-410. MR 806898 (87b:32047)
  • [N] A. Noell and B. Stensønes, Proper holomorphic maps from weakly pseudoconvex domains, preprint.
  • [R] W. Rudin, Function theory in the unit ball of $ {\mathbb{C}^n}$, Springer, New York, 1980. MR 601594 (82i:32002)
  • [S] B. Stensønes, Proper holomorphic mappings from strongly pseudoconvex domains in $ {\mathbb{C}^2}$ to the unit polydisc in $ {\mathbb{C}^3}$, Math. Scand. 65 (1989), 129-139. MR 1051829 (91g:32034)

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Article copyright: © Copyright 1993 American Mathematical Society