The Journal of Geometric Analysis

, Volume 4, Issue 4, pp 467–538 | Cite as

Sufficient conditions for the extension ofC R structures

  • David Catlin


LetM be a smoothC R manifold of dimension 2n − 1 such that at each point, either the Levi form has at least 3 positive eigenvalues or it hasn − 1 negative eigenvalues. IfD is a smoothly bounded subdomain ofM, then there is a smoothly bounded integrable almost complex manifoldX of dimension 2n such thatM is contained in the boundary ofX and such that theC R structure thatM inherits as a subset ofX coincides with the original structure ofM.

Math Subject Classification

(1985 Revision) 32G05 

Key Words and Phrases

Extension ofC R structures integrable almost complex structures Kuranishi embedding problem 


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  1. [1]
    Akahori, A. T. A new approach to the local embedding theorem of CR structures forn ≥ 4.Mem. AMS 366. American Mathematical Society, Providence, RI, 1987.Google Scholar
  2. [2]
    Boutet de Monvel, L. Integration des equations de Cauchy-Riemann induites formelles. Seminaire Goulaouic-Lion-Schwartz (1974–1975).Google Scholar
  3. [3]
    Catlin, D. Extension of CR structures,Proc. Symp. Pure Math. 52(3), 27–34 (1991).MathSciNetGoogle Scholar
  4. [4]
    Catlin, D. A Newlander-Nirenberg theorem for manifolds with boundary.Michigan Math. J. 35, 233–240 (1988).CrossRefMathSciNetMATHGoogle Scholar
  5. [5]
    Cho, S. Extension of complex structures on weakly pseudoconvex compact complex manifolds with boundary.Math. Z. 211, 105–120 (1992).CrossRefMathSciNetMATHGoogle Scholar
  6. [6]
    Folland, G. B., and Kohn, J. J. The Neumann problem for the Cauchy-Riemann complex.Annals of Math. Studies 75. Princeton University Press, Princeton, NJ, 1972.Google Scholar
  7. [7]
    Hill, C. D. What is the notion of a complex manifold with smooth boundary? InProspects of Algebraic Analysis, Sato Anniversary Volume. Academic Press, New York, 1988.Google Scholar
  8. [8]
    Harvey, R., and Lawson, L. B. On the boundaries of complex analytic varieties I, II.Ann. of Math. 102(2), 233–290 (1975);Ann. of Math. 106, 213–238 (1977).CrossRefMathSciNetGoogle Scholar
  9. [9]
    Hörmander, L. L2 estimates and existence theorems for the gd-operator,Acta Math. 113, 89–152 (1965).CrossRefMathSciNetMATHGoogle Scholar
  10. [10]
    Hörmander, L.An Introduction to Complex Analysis in Several Variables. Van Nostrand, Princeton, 1966.MATHGoogle Scholar
  11. [11]
    Jacobowitz, H., and Treves, F. Non-realizable CR structures.Inventiones Math. 66, 231–249 (1982).CrossRefMathSciNetMATHGoogle Scholar
  12. [12]
    Kiremidjian, G. A direct extension method for CR structures.Math. Ann. 242, 1–29 (1976).CrossRefMathSciNetGoogle Scholar
  13. [13]
    Kohn, J. Pseudodifferential operators and hypoellipticity.Proc. Symp. Pure Math. 23 (1973).Google Scholar
  14. [14]
    Kuranishi, M. Strongly pseudoconvex CR structures over small balls.Ann. of Math. 115, 451–500 (1982).CrossRefMathSciNetGoogle Scholar
  15. [15]
    Moser, J. A rapidly convergent iteration method and nonlinear differential equations I.Ann. Scuola Norm. Pisa 20, 265–315 (1966).Google Scholar
  16. [16]
    Nirenberg, L. On elliptic partial differential equations.Ann. Scuola Norm. Sup. Pisa Ser. 3, 13 (1959).Google Scholar
  17. [17]
    Ohsawa, T. Holomorphic embedding of compact s.p.c. manifolds into complex manifolds as real hypersurfaces. InDifferential Geometry of Submanifolds, Lecture Notes in Math., Vol. 1090. Springer-Verlag, Berlin and New York, 1984.Google Scholar
  18. [18]
    Rossi, H. LeBrun’s nonrealizability theorem in higher dimensions.Duke Math. J. 52 (1985).Google Scholar
  19. [19]
    Saint Raymond, X. A simple Nash-Moser implicit function theorem.L’Enseignement Mathematique 35, 217–226 (1989).MathSciNetMATHGoogle Scholar
  20. [20]
    Webster, S. On the proof of Kuranishi’s embedding theorem.Ann. Inst. Henri Poincare 6, 183–207 (1989).MathSciNetMATHGoogle Scholar

Copyright information

© Mathematica Josephina, Inc. 1994

Authors and Affiliations

  • David Catlin
    • 1
  1. 1.Department of MathematicsPurdue UniversityWest Lafayette

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