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Proper holomorphic maps between balls in one co-dimension. (English) Zbl 0699.32014

A proper holomorphic map cannot lower the dimension. When \(f: B^ N\to B^ N\) is proper (here, \(B^ N\) denotes the unit ball in \({\mathbb{C}}^ N)\) then f is an inner automorphism of \(B^ N\) [H. Alexander, Indiana Univ. Math. J. 26, 137-146 (1977; Zbl 0391.32015)], and in particular it extends holomorphically to a neighbourhood of \(B^ N\). Thus for proper \(f: B^ N\to B^ M\) with bad boundary behavior, \(M>N\). Such maps have been constructed by various authors, but always with M very big relative to N. The author constructs a proper holomorphic map \(f: B^ N\to B^{N+1}\) \((N\geq 2)\) that extends continuously to \(\overline{B^ N}\), but that cannot be extended in a \(C^{\infty}\) way to any open, nonempty subset of the boundary.
Reviewer: E.Straube

MSC:

32H35 Proper holomorphic mappings, finiteness theorems
32E35 Global boundary behavior of holomorphic functions of several complex variables

Citations:

Zbl 0391.32015
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References:

[1] Alexander, H., Proper holomorphic mappings inC n ,Indiana Univ. Math. J. 26 (1977), 136–146. · Zbl 0391.32015
[2] Chaumat J. andChollet, A. M., Ensembles pics pourA D),Ann. Inst. Fourier, Grenoble 29 (1979), 171–200. · Zbl 0398.32004
[3] Cima J. A. andSuffridge, T. J., A reflection principal with application to Proper holomorphic mappings,Math. Ann. 265 (1983), 489–500. · Zbl 0525.32021
[4] Cima, J. A. andSuffridge, T. J., Propert holomorphic mappings from the two ball to the three ball,Trans. Amer. Math. Soc. 311 (1989), 227–239.
[5] D’Angelo, J. D., Proper holomorphic maps between balls,Michigan Math. J. 35 (1988), 83–90. · Zbl 0651.32014
[6] D’Angelo, J. D., Polynomial proper holomorphic maps between balls,Duke Math. J. (to appear).
[7] Faran, J., Maps from the two ball to the three ball,Invent. Math. 68 (1982), 441–475. · Zbl 0519.32016
[8] Fornaess J. E. andStensønes Henriksen, B., Peak sets forA k(D).
[9] Forstnerič, F.,Smooth proper holomorphic maps from balls are rational, preprint.
[10] Forstnerič, F., Embedding strictly pseudoconvex domains into balls,Trans. Amer. Math. Soc. 295 (1986), 347–368. · Zbl 0594.32024
[11] Forstnerič, F., Proper holomorphic maps from the ball,Duke Math. J. 53 (1986), 427–440. · Zbl 0603.32019
[12] Globevnik, J., Boundary interpolation by proper holomorphic maps,Math. Z. 194 (1987), 365–373. · Zbl 0611.32021
[13] Hakim, M. andSibony, N., Fonctions holomorphes sur la boule unité deC n ,Invent. Math. 67 (1982), 213–222. · Zbl 0491.32014
[14] Henkin, G. M. andLeiterer, J.,Theory of functions on complex manifolds, Birkhäuser, Basel, 1984.
[15] Kelly, J. L.,General topology, Van Nostrand, Princeton, 1955.
[16] Löw, E., Embeddings and proper holomorphic maps of strictly pseudoconvex domains into polydiscs and balls,Math. Z. 190 (1985), 401–410. · Zbl 0584.32048
[17] Löw, E., A construction of inner functions on the unit ball inC p ,Invent. Math. 67 (1982), 223–229. · Zbl 0528.32006
[18] Rudin, W.,Function theory in the unit ball of C n , Springer, New York, 1980. · Zbl 0495.32001
[19] Stensønes, B.,Proper holomorphic mappings from strongly pseudoconvex domains in C n to the unit polydisc in C n+1 , preprint. · Zbl 0706.32010
[20] Stensønes, B., A zero set forA D) of Hausdorff dimension 2n,Ann. of Math. 125 (1987), 645–659. · Zbl 0628.32017
[21] Wester, S. M.,On mapping an n-ball into (n+1)-ball in a complex space, Pac. J. Math.81 (1979), 267–272. · Zbl 0379.32018
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