Hostname: page-component-77c89778f8-5wvtr Total loading time: 0 Render date: 2024-07-22T00:23:59.122Z Has data issue: false hasContentIssue false
  • English
  • Français

Holomorphic Mappings between Domains in ℂ2

Published online by Cambridge University Press:  20 November 2018

Rasul Shafikov  and
Kaushal Verma
Rasul Shafikov
Affiliation:
Department of Mathematics, The University of Western Ontario, London, On, N6A 5B7 email: shafikov@uwo.ca
Kaushal Verma
Affiliation:
Department of Mathematics, Indian Institute of Science, Bangalore 560 012, India email: kverma@math.iisc.ernet.in
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

An extension theorem for holomorphic mappings between two domains in ${{\mathbb{C}}^{2}}$ is proved under purely local hypotheses.

Keywords

32H4032H40reflection principleSegre varieties
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 64 , Issue 2 , 16 April 2012 , pp. 429 - 454
Copyright
Copyright © Canadian Mathematical Society 2012

References

[1] Baouendi, M. S., Ebenfelt, P. and Rothschild, L. P., Real Submanifolds in Complex Space and Their Mappings. Princeton Mathematical Series 47. Princeton University Press, Princeton, NJ, 1999.Google Scholar
[2] Berteloot, F. and Sukhov, A. On the continuous extension of holomorphic correspondences. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 24(1997), no. 4, 747–766.Google Scholar
[3] Chirka, E., Regularity of the boundaries of analytic sets. Mat. Sb. (N. S.) 117(159)(1982), no. 3, 291–336, 431.Google Scholar
[4] Chirka, E., Complex Analytic Sets. Mathematics and its Applications (Soviet Series) 46. Kluwer Academic Publishers Group, Dordrecht, 1989.Google Scholar
[5] Coupet, B., Pinchuk, S. and Sukhov, A. On boundary rigidity and regularity of holomorphic mappings. Internat. J. Math. 7(1996), no. 5, 617–643. http://dx.doi.org/10.1142/S0129167X96000335 Google Scholar
[6] Diederich, K. and Fornaess, J. E., Proper holomorphic mappings between real-analytic pseudoconvex domains in Cn. Math. Ann. 282(1988), no. 4, 681–700. http://dx.doi.org/10.1007/BF01462892 Google Scholar
[7] Diederich, K. and Fornaess, J. E., Biholomorphic mappings between certain real analytic domains in C2. Math. Ann. 245(1979), no. 3, 255–272. http://dx.doi.org/10.1007/BF01673510 Google Scholar
[8] Diederich, K., Fornaess, J. E., and Ye, Z. Biholomorphisms in dimension 2. J. Geom. Anal. 4(1994), no. 4, 539–552. http://dx.doi.org/10.1007/BF01896407 Google Scholar
[9] Diederich, K. and Pinchuk, S. Proper holomorphic maps in dimension 2 extend. Indiana Univ. Math. J. 44(1995), no. 4, 1089–1126.Google Scholar
[10] Diederich, K. and Pinchuk, S., Regularity of continuous CR maps in arbitrary dimension. Michigan Math. J. 51(2003), no. 1, 111–140. http://dx.doi.org/10.1307/mmj/1049832896 Google Scholar
[11] Diederich, K. and Pinchuk, S., Analytic sets extending the graphs of holomorphic mappings. J. Geom. Anal. 14(2004), no. 2, 231–239.Google Scholar
[12] Diederich, K. and Webster, S. M., A reflection principle for degenerate real hypersurfaces. Duke Math. J. 47(1980), no. 4, 835–843. http://dx.doi.org/10.1215/S0012-7094-80-04749-3 Google Scholar
[13] Fornaess, J. E. and Low, E. Proper holomorphic mappings. Math. Scand. 58(1986), no. 2, 311–322.Google Scholar
[14] Fornaess, J. E. and Sibony, N. Construction of P. S.H. unctions on weakly pseudoconvex domains. Duke Math. J. 58(1989), no. 3, 633–655. http://dx.doi.org/10.1215/S0012-7094-89-05830-4 Google Scholar
[15] Hironaka, H., Subanalytic sets. In: Number Theory, Algebraic Geometry and Commutative Algebra, in Honor of Yasuo Akizuki. Kinokuniya, Tokyo, 1973, pp. 453–493.Google Scholar
[16] Łojasiewicz, S., Introduction to Complex Analytic Geometry. Birkhäuser-Verlag, Basel, 1991.Google Scholar
[17] Narasimhan, R., Introduction to the Theory of Analytic Spaces. Lecture Notes in Mathematics 25. Springer-Verlag, Berlin, 1966.Google Scholar
[18] Pinchuk, S. and Tsyganov, S. Smoothness of CR-mappings between strictly pseudoconvex hypersurfaces. Izv. Akad. Nauk SSSR Ser. Mat. 53(1989), no. 5, 1120–1129, 1136; translation in Math. USSR-Izv. 35(1990), no. 2, 457–467.Google Scholar
[19] Rudin, W., Function Theory in the Unit Ball of Cn. Grundlehren der Mathematischen Wissenschaften 241. Springer-Verlag, New York, 1980.Google Scholar
[20] Sibony, N., Some aspects of weakly pseudoconvex domains. In: Several Complex Variables and Complex Geometry, Part 1. Proc. Sympos. Pure Math. 52. American Mathematical Society, Providence, RI, 1991, pp. 199–231.Google Scholar
[21] Sukhov, A., On the boundary regularity of holomorphic mappings. Mat. Sb. 185(1994), no. 12, 131–142; translation in Russian Acad. Sci. Sb. Math. 83(1995), no. 2, 541–551Google Scholar
[22] Shafikov, R., Analytic continuation of germs of holomorphic mappings between real hypersurfaces in Cn. Michigan Math. J. 47(2000), 133–149. http://dx.doi.org/10.1307/mmj/1030374673 Google Scholar
[23] Shafikov, R. and Verma, K. A local extension theorem for proper holomorphic mappings in C2. J. Geom. Anal. 13(2003), no. 4, 697–714.Google Scholar
[24] Trépreau, J.-M., Sur le prolongement holomorphe des fonctions C-R définés sur une hypersurface réelle de classe C2 dans Cn. Invent. Math. 83(1986), no. 3, 583–592. http://dx.doi.org/10.1007/BF01394424 Google Scholar
[25] Verma, K., Boundary regularity of correspondences in C2. Math. Z. 231(1999), no. 2, 253–299. http://dx.doi.org/10.1007/PL00004728Google Scholar