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Holomorphic Extension of CR Functions: A Survey

  • Jean-Marie Trépreau
Conference paper
Part of the Progress in Nonlinear Differential Equations and Their Applications book series (PNLDE, volume 21)

Abstract

A CR function on a generic real submanifold of Cn is a solution of the induced Cauchy-Riemann system; for example, traces of holomorphic functions are CR. This lecture will be a survey, intended for the non-specialist, of the results obtained during the last ten years on the question of the holomorphic extendibility of CR functions. Emphasis will be put on the method of the complex disc, the Bishop equation, and Tumanov’s theorem on the holomorphic extendibility into a wedge from minimal manifolds. The basic notions about CR functions will be introduced and several important results will be presented.

Keywords

Holomorphic Function Finite Type Extension Property Real Hypersurface Holomorphic Extension 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    A. Andreotti and C. D. Hill, E. E. Levi convexity and the Hans Lewy problem, Ann. Scuola Norm. Sup. Pisa 26 (1972), 325–363.MathSciNetMATHGoogle Scholar
  2. 2.
    M. S. Baouendi, C. H. Chang and F. Treves, Microlocal hypo-analyticity and extension of CR functions, J. Diff. Geom. 18 (1983), 331–391.MathSciNetMATHGoogle Scholar
  3. 3.
    M. S. Baouendi and L. P. Rothschild, Cauchy-Riemann functions on manifolds of higher codimension in complex space, Invent. Math. 101 (1990), 45–56.MathSciNetMATHGoogle Scholar
  4. 4.
    M. S. Baouendi, L. P. Rothschild and J.-M. Trépreau, On the geometry of analytic discs attached to real manifolds, J. Diff. Geom. 39 (1994), 379–405.MathSciNetMATHGoogle Scholar
  5. 5.
    M. S. Baouendi and F. Treves, A property of the functions and distributions annihilated by a locally integrable system of complex vector fields, Ann. of Math. 113 (1981), 387–421.MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    M. S. Baouendi and F. Treves, About the holomorphic extension of CR functions on real hypersurfaces in complex space, Duke Math. J. 51 (1984), 77–107.MathSciNetMATHGoogle Scholar
  7. 7.
    E. Bedford and B. Gaveau, Envelopes of holomorphy of certain two-spheres in C2, Amer. J. Math. 105 (1983), 975–1009.MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    E. Bedford and Klingenberg, On the envelope of holornorphy of a 2-sphere in C 2, J. Amer. Math. Soc. 4 (1991), 623–646.MathSciNetGoogle Scholar
  9. 9.
    E. Bishop, Differentiable manifolds in complex Euclidean space, Duke Math. J. 32 (1965), 1–22.MATHGoogle Scholar
  10. 10.
    J. E. Fornss and C. Rea, Local holomorphic extendibility and non-extendibility of CR-functions on smooth boundaries, Ann. Scuola Norm. Sup. Pisa 12 (1985), 491–502.Google Scholar
  11. 11.
    F. Forstneric, Analytic discs with boundary in a maximal totally real submanifold of C“, Ann. Inst. Fourier, Grenoble 37 (1987), 1–44.MATHGoogle Scholar
  12. 12.
    J. Globevnik, Perturbation of analytic discs along maximal real submanifolds of C“, Math. Z. 217 (1994), 287–316.MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    M. Gromov, Pseudo-holomorphic curves in symplectic manifolds, Invent. Math. 82 (1985), 307–347.MATHGoogle Scholar
  14. 14.
    N. Hanges and F. Treves, On the local holomorphic extension of CR functions, Astérisque 217 (1993), 119–137, Colloque Analyse Complexe et Géométrie, Marseille 1992.Google Scholar
  15. 15.
    L. Hörmander, An introduction to complex analysis in several variables, North-Holland Publ. Co., Amsterdam, London, 1973.Google Scholar
  16. 16.
    B. Jöricke, Deformation of CR manifolds, minimal points and CR manifolds with the microlocal extension property, J. Geom. Anal. (to appear).Google Scholar
  17. 17.
    J. J. Kohn and L. Nirenberg, A pseudoconvex domain not admitting a holomorphic support function, Math. Ann. 201 (1973), 265–268.MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    H. Lewy, On the local character of the solution of an atypical differential equation in three variables and a related problem for regular functions of two complex variables, Ann. of Math. 64 (1956), 514–522.MathSciNetCrossRefGoogle Scholar
  19. 19.
    J. Merker, Global minimality of generic manifolds and holomorphic extendibility of CR functions, Int. Math. Res. Not. 8 (1994), 329–342.Google Scholar
  20. 20.
    C. Rea, Prolongement holomorphe des fonctions CR, conditions suffisantes, Notes aux C. R. Acad. Sci. Paris 297 série 1 (1983), 163–165.Google Scholar
  21. 21.
    M. Sato, T. Kawai and M. Kashiwara, Hyperfunctions and pseudodifferential equations, Springer Lecture Notes in Math. 287 (1973), 265–529.MathSciNetGoogle Scholar
  22. 22.
    J. Sjöstrand, Singularités analytiques microlocales, Astérisque 95 (1982), 1–166. 23, The FBI-transform for CR submanifolds of C“, Prépublications Math. d’Orsay, Université Paris-Sud 1982.Google Scholar
  23. 23.
    H. J. Sussmann, Orbits of families of vector fields and integrability of distributions, Trans. Amer. Math. Soc. 180 (1973), 171–187.MathSciNetCrossRefGoogle Scholar
  24. 24.
    J.-M. Trépreau, Sur le prolongement holomorphe des functions CR définies sur une hypersurface réelle de classe C2 dans C“, Invent. Math. 83 (1986), 583–592.MATHGoogle Scholar
  25. 25.
    J.-M. Trépreau, Sur la propagation des singularités dans les variétés CR, Bull. Soc. Math. France 118 (1990), 403–450.MathSciNetMATHGoogle Scholar
  26. 26.
    J.-M. Trépreau, On the extension of holomorphic functions across a real hypersurface, Math. Z. 211 (1992), 93–103.MathSciNetCrossRefGoogle Scholar
  27. 27.
    F. Treves, Approximation and representation of functions and distributions annihilated by a system of complex vector fields, Ecole Polytechnique, Centre de Math., Palaiseau 1981.Google Scholar
  28. 28.
    A. E. Tumanov, Extension of CR functions into a wedge from a manifold of finite type, Mat. Sbornik 136 (1988), 129–140.Google Scholar
  29. 29.
    A. E. Tumanov, Extension of CR-functions into a wedge, Mat. Sbornik 181 (1990), 951–964.Google Scholar
  30. 30.
    A. E. Tumanov, Connections and propagation of analyticity for CR functions, Duke Math. J. 73 (1994), 1–24.Google Scholar
  31. 31.
    A. E. Tumanov, Connections and propagation of analyticity for CR functions, Duke Math. J. 73 (1994), 1–24.MathSciNetMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1996

Authors and Affiliations

  • Jean-Marie Trépreau
    • 1
  1. 1.Institut de MathématiquesUniversité de Paris VIParis Cedex 05France

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