Skip to main content
SpringerLink
Log in

Remarks on the rank properties of formal CR maps

  • Published:
Science in China Series A: Mathematics Aims and scope Submit manuscript

Abstract

We prove several new transversality results for formal CR maps between formal real hypersurfaces in complex space. Both cases of finite and infinite type hypersurfaces are tackled in this note.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Fornæss J E. Biholomorphic mappings between weakly pseudoconvex domains. Pacific J Math, 1978, 74: 63–65

    MathSciNet  MATH  Google Scholar 

  2. Baouendi M S, Rothschild L P. Germs of CR maps between real-analytic hypersurfaces. Invent Math, 1988, 93: 481–500

    Article  MathSciNet  MATH  Google Scholar 

  3. Diederich K, Fornæss J E. Proper holomorphic mappings between real analytic pseudoconvex domains in ℂn. Math Ann, 1988, 282: 681–700

    Article  MathSciNet  MATH  Google Scholar 

  4. Huang X. Schwarz reflection principle in complex spaces of dimension two. Comm Part Diff Eq, 1996, 21(11–12): 1781–1829.

    MATH  Google Scholar 

  5. Baouendi M S, Rothschild L P. A generalized complex Hopf lemma and its application to CR mappings. Invent Math, 1993, 11: 331–348

    Article  MathSciNet  Google Scholar 

  6. Baouendi M S, Huang X. Nonvanishing of the differential of holomorphic mappings at boundary points. Math Res Lett, 1995, 2: 737–750

    MathSciNet  MATH  Google Scholar 

  7. Huang X, Pan Y. Proper holomorphic mappings between real analytic domains in ℂn. Duke Math J, 1996, 82: 437–446

    Article  MathSciNet  MATH  Google Scholar 

  8. Chirka E, Rea C, Normal and tangent ranks of CR mappings. Duke Math J. 1998, 94: 417–431

    Article  MathSciNet  Google Scholar 

  9. Kohn J J. Boundary behavior of \(\bar \partial \) on weakly pseudo-convex manifolds of dimension two. J Differential Geom, 1972, 6: 523–542

    MathSciNet  MATH  Google Scholar 

  10. Bloom T, Graham I. On type conditions for generic real submanifolds of ℂn. Invent Math, 1977, 40: 217–243

    Article  MathSciNet  Google Scholar 

  11. Baouendi M S, Rothschild L P. Geometric properties of mappings between hypersurfaces in complex space. J Differential Geom, 1990, 31: 473–499

    MathSciNet  MATH  Google Scholar 

  12. Baouendi M S, Rothschild L P. A general reflection principle in ℂ2. J Funct Anal, 1991, 99: 409–442

    Article  MathSciNet  MATH  Google Scholar 

  13. Meylan F. A reflection principle for a class of hypersurfaces and mappings. Pacific J Math, 1995, 169: 135–160

    MathSciNet  MATH  Google Scholar 

  14. Baouendi M S, Ebenfelt P. Rothschild L P, Real Submanifolds in Complex Space and Their Mappings. Princeton Math Series 47: Princeton: Princeton Univ Press, 1999

    MATH  Google Scholar 

  15. Stanton N, Infinitesimal CR automorphisms of real hypersurfaces. Amer J Math, 1996, 118: 209–233

    MathSciNet  MATH  Google Scholar 

  16. Baouendi M S, Ebenfelt P, Rothschild L P. Convergence and finite determination of formal CR mappings. J Amer Math Soc, 2000, 13: 697–723

    Article  MathSciNet  MATH  Google Scholar 

  17. Kowalski R T. A hypersurface in ℂ2 whose stability group is not determined by 2-jets. Proc Amer Math Soc, 2002, 130: 3679–3686

    Article  MathSciNet  MATH  Google Scholar 

  18. Zaitsev D. Unique jet determination of local CR-maps by their jets: a survey. Atti Accad Naz Lincei Cl Sci Fis Mat Natur Rend Lincei (9) Mat Appl, 2002, 13: 295–305

    MathSciNet  MATH  Google Scholar 

  19. Kowalski S T. Rational jet dependence of formal equivalences between real-analytic hypersurfaces in ℂ2. Pacific J Math, 2005, 220: 107–139

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Fakultät für Mathematik, Universität Wien, Nordbergstrasse 15, A-1090, Wien, Österreich

    Lamel Bernhard

  2. Laboratoire de Mathématiques Raphaël Salem, UMR 6085 CNRS, Université de Rouen, Avenue de l’Université, B.P. 12, 76801 Saint Etienne du Rouvray, France

    Mir Nordine

Authors
  1. Lamel Bernhard
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mir Nordine
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Lamel Bernhard.

Additional information

Dedicated to Professor Sheng GONG on the occasion of his 75th birthday

Rights and permissions

Reprints and permissions

About this article

Cite this article

Lamel, B., Mir, N. Remarks on the rank properties of formal CR maps. SCI CHINA SER A 49, 1477–1490 (2006). https://doi.org/10.1007/s11425-006-2047-8

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11425-006-2047-8

Keywords

MSC(2000)

Search

Navigation