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Analytic sets and the boundary regularity of CR mappings


Authors: Sergey Pinchuk and Kaushal Verma
Journal: Proc. Amer. Math. Soc. 129 (2001), 2623-2632
MSC (1991): Primary 32V10; Secondary 32V25
DOI: https://doi.org/10.1090/S0002-9939-01-05970-6
Published electronically: March 15, 2001
MathSciNet review: 1838785
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Abstract:

It is shown that if a continuous CR mapping between smooth real analytic hypersurfaces of finite type in ${\mathbf C}^n$ extends as an analytic set, then it extends as a holomorphic mapping.


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  • [BBR] Baouendi, M. S., Bell, S., Rothschild, L. P.: Mappings of three-dimensional CR manifolds and their holomorphic extension, Duke Math. J. 56 (1988), 503-530. MR 90a:32035
  • [BJT] Baouendi, M. S., Jacobowitz, H., Treves, F.: On the analyticity of CR mappings, Ann. Math. 122 (1985), 365-400. MR 87f:32044
  • [BR] Baouendi, M. S., Rothschild, L. P.: Remarks on the generic rank of a CR mapping, J. Geom. Anal. 2 No. 1 (1992), 1-9. MR 92m:32025
  • [BB] Bedford, E., Bell, S.: Extension of proper holomorphic mappings past the boundary, Manuscr. Math. 50 (1985), 1-10. MR 86h:32041
  • [BS] Berteloot, F., Sukhov, A.: On the continuous extension of holomorphic correspondences, Ann. Sc. Norm. Super. Pisa, Series 4 24 No. 4 (1997), 747-766. MR 2000d:32029
  • [C] Chirka, E. M.: Complex analytic sets, Kluwer, Dordrecht (1989). MR 92b:32016
  • [CP] Coupet, B., Pinchuk, S.: Holomorphic equivalence problem for weighted homogeneous rigid domains in ${\mathbf C}^{n + 1}$, Collection of papers dedicated to B. V. Shabat, Ed. E. M. Chirka, Farsis, Moscow (1997), 111-126.
  • [DF1] Diederich, K., Fornaess, J. E.: Proper holomorphic mappings between real analytic pseudoconvex domains in ${\mathbf C}^n$, Math. Ann. 282 (1988), 681-700. MR 89m:32045
  • [DF2] Diederich, K., Fornaess, J. E.: Applications holomorphes propres entre domaines a bord analytique reel, C. R. Acad. Sci. Paris Ser. I Math. 307 (1988), 321-324. MR 89i:32052
  • [DP1] Diederich, K., Pinchuk, S.: Proper holomorphic maps in dimension 2 extend, Indiana Univ. Math. J. 44 (1995), 1089-1126. MR 97g:32031
  • [DP2] Diederich, K., Pinchuk, S.: Reflection principle in higher dimensions, Doc. Math. J. Extra Volume ICM (1998) Part II, 703-712. MR 2000d:32031
  • [DW] Diederich, K., Webster, S.: A reflection principle for degenerate real hypersurfaces, Duke Math. J. 47 (1980), 835-845. MR 82j:32046
  • [H] Huang, X.: Schwarz reflection principle in complex spaces of dimension two, Comm. Partial Differential Equations 21 (1996), 1781-1828. MR 97m:32043
  • [L] Lojasiewicz, S.: Introduction to complex analytic geometry, Birkhauser, Boston (1991). MR 92g:32002
  • [P] Pinchuk, S.: On the analytic continuation of holomorphic mappings, Math. USSR Sb. (N.S.) 98 (1975), 416-435.
  • [PT] Pinchuk, S., Tsyganov, Sh.: CR mappings of strictly pseudoconvex hypersurfaces, Izv. Akad. Nauk SSSR 53 (1989), 1020-1029. MR 90j:32022
  • [S] Shabat, B. V.: Introduction to complex analysis, Part II, Translations of Mathematical Monographs 110, AMS (1992). MR 93g:32001
  • [Sh] Shafikov, R.: On the analytic continuation of holomorphic mappings on real analytic hypersurfaces, preprint 1998.
  • [T] Trepreau, J. M.: Sur le prolongement holomorphe des fonctions CR definies sur une hypersurface reele de classe $C^2$ dans ${\mathbf C}^n$, Invent. Math. 83 (1986), 583-592. MR 87f:32035
  • [V] Verma, K.: Boundary regularity of correspondences in ${\mathbf C}^2$, Math. Z. 231 No. 2 (1999), 253-299. MR 2000g:32021
  • [Vl] Vladimirov, V.: Methods of the theory of functions of many complex variables, M.I.T. Press, 1966. MR 34:1551

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Additional Information

Sergey Pinchuk
Affiliation: Department of Mathematics, Indiana University, Bloomington, Indiana 47401
Email: pinchuk@indiana.edu

Kaushal Verma
Affiliation: Department of Mathematics, Syracuse University, Syracuse, New York 13244
Address at time of publication: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109
Email: kkverma@syr.edu, kverma@math.lsa.umich.edu

DOI: https://doi.org/10.1090/S0002-9939-01-05970-6
Keywords: CR mappings, correspondences
Received by editor(s): December 28, 1999
Published electronically: March 15, 2001
Additional Notes: The first author’s research was supported in part by a fund from the NSF
Communicated by: Steven R. Bell
Article copyright: © Copyright 2001 American Mathematical Society

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