Boundary regularity of correspondences in ℂn

  • Rasul Shafikov
  • Kaushal Verma


LetM, M′ be smooth, real analytic hypersurfaces of finite type in ℂn and\(\hat f\) a holomorphic correspondence (not necessarily proper) that is defined on one side ofM, extends continuously up toM and mapsM to M′. It is shown that\(\hat f\) must extend acrossM as a locally proper holomorphic correspondence. This is a version for correspondences of the Diederich-Pinchuk extension result for CR maps.


Correspondences Segre varieties 


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  1. [BR]
    Baouendi M and Rothschild L, Germs of CR maps between real analytic hypersurfaces,Invent. Math. 93 (1988) 481–500zbMATHCrossRefMathSciNetGoogle Scholar
  2. [B]
    Bedford E, Proper holomorphic mappings from domains with real analytic boundary,Am. J. Math. 106 (1984) 745–760zbMATHCrossRefMathSciNetGoogle Scholar
  3. [BS]
    Berteloot F and Sukhov A, On the continuous extension of holomorphic correspondences,Ann. Scuola Norm. Sup. Pisa Cl. Sci. 24 (1997) 747–766zbMATHMathSciNetGoogle Scholar
  4. [C]
    Chirka E M, Complex analytic sets (Dordrecht: Kluwer) (1990)Google Scholar
  5. [DF]
    Diederich K and Fornaess J E, Proper holomorphic mappings between real analytic pseudoconvex domains in Cn,Math. Ann. 282 (1988) 681–700zbMATHCrossRefMathSciNetGoogle Scholar
  6. [DP1]
    Diederich K and Pinchuk S, Proper holomorphic maps in dimension 2 extend,Indiana Univ. Math. J. 44 (1995) 1089–1126zbMATHCrossRefMathSciNetGoogle Scholar
  7. [DP2]
    Diederich K and Pinchuk S, Reflection principle in higher dimensions,Doc. Math. J. Extra Volume ICM (1998) PartII, pp. 703–712Google Scholar
  8. [DP3]
    Diederich K and Pinchuk S, Regularity of continuous CR maps in arbitrary dimension,Mich. Math. J. 51(1) (2003) 111–140zbMATHCrossRefMathSciNetGoogle Scholar
  9. [DP4]
    Diederich K and Pinchuk S, Analytic sets extending the graphs of holomorphic mappings,J. Geom. Anal. 14(2) (2004) 231–239zbMATHMathSciNetGoogle Scholar
  10. [DW]
    Diederich K and Webster S, A reflection principle for degenerate real hypersurfaces,Duke Math. J. 47 (1980) 835–845zbMATHCrossRefMathSciNetGoogle Scholar
  11. [S1]
    Shafikov R, Analytic continuation of germs of holomorphic mappings between real hypersurfaces in ℂn,Mich. Math. J. 47(1) (2001) 133–149CrossRefMathSciNetGoogle Scholar
  12. [S2]
    Shafikov R, On boundary regularity of proper holomorphic mappings,Math. Z. 242(3) (2002) 517–528zbMATHCrossRefMathSciNetGoogle Scholar
  13. [SV]
    Shafikov R and Verma K, A local extension theorem for proper holomorphic mappings in ℂ2, J.Geom. Anal. 13(4) (2003) 697–714zbMATHMathSciNetGoogle Scholar
  14. [V]
    Verma K, Boundary regularity of correspondences in C2,Math. Z. 231(2) (1999) 253–299zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Indian Academy of Sciences 2006

Authors and Affiliations

  • Rasul Shafikov
    • 1
  • Kaushal Verma
    • 2
  1. 1.Department of Mathematics, Middlesex CollegeUniversity of Western OntarioLondonUSA
  2. 2.Department of MathematicsIndian Institute of ScienceBangaloreIndia

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