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Proper holomorphic maps from domains in ${ℂ}^{2}$ with transverse circle action. (English) Zbl 1143.32011

Let $T={S}^{1}$ denote the torus and let ${\Omega }$ be a bounded connected open subset of ${ℂ}^{2}$, which is pseudoconvex, of finite type and with smooth three dimensional boundary.

In this paper the authors consider proper holomorphic mappings between pseudoconvex regions of ${ℂ}^{2}$ and they study transverse actions in relation with the branch locus. Recall that classes of domains admitting a $T$-action are for instance Hartogs domains, Reinhardt and quasi-circular domains.

##### MSC:
 32H35 Proper mappings, finiteness theorems 32T25 Finite type domains
##### References:
 [1] Barrett, D., Regularity of the Bergman projection on domains with transverse symmetries, Math. Ann., 258(4), 1981/82, 441–446. MR0650948 (83i:32032) · Zbl 0486.32015 · doi:10.1007/BF01453977 [2] Coupet, B., Pan, Y. and Sukhov, A., On proper holomorphic mappings from domains with T-action, Nagoya Math. J., 154, 1999, 57–72. MR1689172 (2000b:32036) [3] Pan, Y., Proper holomorphic self-mappings of Reinhardt domains, Math. Z., 208(2), 1991, 289–295. MR1128711 (93f:32029) · Zbl 0727.32011 · doi:10.1007/BF02571526 [4] Coupet B., Pan, Y., and Sukhov, A., Proper holomorphic self-maps of quasi-circular domains in $ℂ$2, Nagoya Math. J., 164, 2001, 1–16. MR1869091 (2002j:32014) [5] D’Angelo, J., Several Complex Variables and the Geometry of Real Hypersurfaces, CRC Press, Boca Raton, 1993. MR1224231 (94i:32022) [6] Bedford, E., Action of the automorphisms of a smooth domain in $ℂ$ n , Proc. Amer. Math. Soc., 93(2), 1985, 232–234. MR0770527 (86e:32029) [7] Bell, S., Boundary behavior of proper holomorphic mappings between non-pseudoconvex domains, Amer. J. Math., 106(3), 1984, 639–643. MR0745144 (86a:32054) · Zbl 0552.32019 · doi:10.2307/2374288 [8] Bell, S. and Catlin, D., Boundary regularity of proper holomorphic mappings, Duke Math. J., 49, 1982, 385–396. MR0659947 (84b:32037a) · Zbl 0486.32014 · doi:10.1215/S0012-7094-82-04924-9 [9] Diederich, K. and Fornæss, J. E., Boundary regularity of proper holomorphic mappings, Invent. Math., 67(3), 1982, 363–384. MR0664111 (84b:32037b) · Zbl 0501.32010 · doi:10.1007/BF01398927 [10] Range, R. M., Holomorphic Functions and Integral Representations in Several Complex Variables, GTM 108, Springer, New York, 1986. MR0847923 (87i:32001) [11] Rudin, W., Function Theory in the Unit Ball of $ℂ$ n , GMW 241, Springer-Verlag, New York, 1980. MR0601594 (82i:32002) [12] Bedford, E., Proper holomorphic mappings, Bull. Amer. Math. Soc. (N.S.), 10(2), 1984, 157–175. MR073 3691 (85b:32041) · Zbl 0534.32009 · doi:10.1090/S0273-0979-1984-15235-2 [13] Bell, S., Local boundary behavior of proper holomorphic mappings, Complex Analysis of Several Variables, Proc. Sympos. Pure Math., 41, A. M. S., Providence, 1984, 1–7. MR0740867 (85j:32043) [14] Bedford, E. and Fornæss, J. E., A construction of peak functions on weakly pseudoconvex domains, Annals of Math., 107(3), 1978, 555–568. MR0492400 (58 #11520) · Zbl 0392.32004 · doi:10.2307/1971128 [15] Fornæss, J. E. and McNeal, J., A construction of peak functions on some finite type domains, Amer. J. Math., 116(3), 1994, 737–755. MR1277453 (95j:32023) · Zbl 0809.32005 · doi:10.2307/2374998 [16] Diederich, K. and Fornæss, J. E., Proper holomorphic images of strictly pseudoconvex domains, Math. Ann., 259(2), 1982, 279–286. MR0656667 (83g:32026) · Zbl 0486.32013 · doi:10.1007/BF01457314 [17] Grauert, H. and Remmert, R., Coherent Analytic Sheaves, GMW 265, Springer, Berlin, 1984. MR0755331 (86a:32001) [18] Bedford, E., Proper holomorphic mappings from strongly pseudoconvex domains, Duke Math. J., 49(2), 1982, 477–484. MR0659949 (84b:32036) · Zbl 0498.32011 · doi:10.1215/S0012-7094-82-04926-2 [19] Huang, X. and Ji, S., Global holomorphic extension of a local map and a Riemann mapping theorem for algebraic domains, Math. Res. Lett., 5(1–2), 1998, 247–260. MR1617897 (99d:32013) [20] Boothby, W., An Introduction to Differentiable Manifolds and Riemannian Geometry, Second Edition, Pure and Applied Math., 120, Academic Press, Boston, 1986. MR0861409 (87k:58001) [21] Diederich, K. and Fornæss, J. E., Pseudoconvex domains: bounded strictly plurisubharmonic exhaustion functions, Invent. Math., 39(2), 1977, 129–141. MR0437806 (55 #10728) · Zbl 0353.32025 · doi:10.1007/BF01390105 [22] Pinčuk, S. I., A boundary uniqueness theorem for holomorphic functions of several complex variables, Mat. Zametki, 15(2), 1974, 205–212; English transl., Math. Notes, 15(1–2), 116–120. MR0350065 (50 #2558) [23] Baouendi, M. S., Rothschild, L. P. and Trèves, F., CR structures with group action and extendability of CR functions, Invent. Math., 82(2), 1985, 359–396. MR0809720 (87i:32028) · Zbl 0598.32019 · doi:10.1007/BF01388808 [24] Chirka, E. M., Introduction to the geometry of CR manifolds, Uspekhi Mat. Nauk, 46(1), 1991, 81–164, Trans. in Russian Math. Surveys, 46(1), 1991, 95–197. MR1109037 (92m:32012) [25] Tanaka, N., On the pseudo-conformal geometry of hypersurfaces of the space of n complex variables, J. Math. Soc. Japan, 14, 1962. 397–429. MR0145555 (26 #3086) · Zbl 0113.06303 · doi:10.2969/jmsj/01440397 [26] Bell, S. and Catlin, D., Regularity of CR mappings, Math. Z., 199(3), 1988, 357–368. MR0961816 (89i:32028) · Zbl 0639.32011 · doi:10.1007/BF01159784