Skip to main content
SpringerLink
Log in

Boundaries of singularity sets, removable singularities, and CR-invariant subsets of CR-manifolds

  • Published:
The Journal of Geometric Analysis Aims and scope Submit manuscript

Abstract

Let Ω be a bounded strictly pseudoconvex domain in ℂn, n ≥ 3, with boundary ∂Ω, of class C2. A compact subset K is called removable if any analytic function in a suitable small neighborhood of ∂Ω K extends to an analytic function in Ω. We obtain sufficient conditions for removability in geometric terms under the condition that K is contained in a generic C2 -submanifold M of co-dimension one in ∂Ω. The result uses information on the global geometry of the decomposition of a CR-manifold into CR-orbits, which may be of some independent interest. The minimal obstructions for removability contained in M are compact sets K of two kinds. Either K is the boundary of a complex variety of co-dimension one in Ω or it is an exceptional minimal CR-invariant subset of M, which is a certain analog of exceptional minimal sets in co-dimension one foliations. It is shown by an example that the latter possibility may occur as a nonremovable singularity set.

Further examples show that the germ of envelopes of holomorphy of neighborhoods of ∞Ω K for K ⊂ M may be multisheeted. A couple of open problems are discussed.

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. Airapetian, R.A. Continuation of CR-functions from piecewise smooth CR-manifolds,Mat. Sb.,134(176), 108–118, (1987).

    Google Scholar 

  2. Anderson, J.T. and Cima, J.A. Removable singularities for Lp CR-functions,Mich. Math. J.,41, 111–119, (1994).

    Article  MathSciNet  MATH  Google Scholar 

  3. Baouendi, M.S. and Treves, F. A property of the functions and distributions annihilated by a locally integrable system of complex vector fields,Ann. Math.,113, 387–421, (1981).

    Article  MathSciNet  Google Scholar 

  4. Berndtsson, B. and Ransford, T.J. Analytic multifunctions, the\(\bar \partial \)-equation, and a proof of the corona theorem,Pacif. J. Math.,124, 57–72, (1986).

    MathSciNet  MATH  Google Scholar 

  5. Čirka, E.M.Complex Analytic Sets (Russian). Moscow, Nauka 1985; English translation, Kluwer, Dordrecht, 1989.

    Google Scholar 

  6. Čirka, E.M. and Stout, E.L. Removable singularities in the boundary, Contributions to Complex Analysis and Analytic Geometry, dedicated to P. Dolbeault, Vieweg, Ed., H. Skoda, J.-M. Trépreau, 43–104.

  7. Dimca, A. Personal communication.

  8. Duval, J. Surfaces convexes dans un bord pseudoconvex,Astérisque,217(6), 103–118, (1993).

    MathSciNet  Google Scholar 

  9. Duval, J. and Sibony, N. Polynomial convexity, rational convexity and currents,Duke Math. J.,79, 487–519 (1995).

    Article  MathSciNet  MATH  Google Scholar 

  10. Fornæss, J.E. and Sibnony, N. Plurisubharmonic functions on ring domains,Lecture Notes in Math.,1268, 111–120, Springer, Berlin, 1987.

    Google Scholar 

  11. Forstnerič, F. On the boundary regularity of proper mappings,Annali Scuola Normale Sup. Pisa, Ser., IV, XIII, 109–128 (1986).

    Google Scholar 

  12. Forstnerič, F. and Stout, E.L. A new class of polynomially convex sets,Ark. för Mat.,29, 51–62 (1991).

    Article  MATH  Google Scholar 

  13. Ghys, E. Personal communication.

  14. Golubitski, M. and Guillermin, V.Stable Mappings and Their Singularities, Springer, New York, 1973.

    Google Scholar 

  15. Grauert, H. On Levi’s problem,Ann. Math.,68(2), 460–472, (1958).

    Article  MathSciNet  Google Scholar 

  16. Greenfield, S.J. Cauchy-Riemann equations in several variables,Ann. Scuola Norm. Sup. Pisa,22, 275–314, (1968).

    MathSciNet  MATH  Google Scholar 

  17. Hartmann, Ph.Ordinary Differential Equations, John Wiley & Sons, New York, 1964.

    Google Scholar 

  18. Harvey, F.R. and Lawson, H.B. On boundaries of complex analytic varieties, I,Ann. Math.,102, 223–290, (1975).

    Article  MathSciNet  Google Scholar 

  19. Hector, G. and Hirsch, U.Introduction to the Geometry of Foliations, Part A & Part B, Fr. Vieweg & Sohn, Braunschweig/Wiesbaden, 1983.

    Google Scholar 

  20. Hörmander, L.An Introduction to Complex Analysis in Several Variables, D. van Nostrand, Princeton, NJ, 1966.

    MATH  Google Scholar 

  21. Jöricke, B. Removable singularities of CR-functions,Ark. för Mat.,26, 117–143, (1988).

    Article  MATH  Google Scholar 

  22. Jöricke, B. Envelopes of holomorphy and CR-invariant subsets of CR-manifolds,C.R. Acad. Sc. Paris,315, Sér. I, 407–411, (1992).

    MATH  Google Scholar 

  23. Jöricke, B. Some remarks concerning holomorphically convex hulls and envelopes of holomorphy,Math. Z.,218, 143–157, (1995).

    Article  MathSciNet  MATH  Google Scholar 

  24. Jöricke, B. Deformation of CR-manifolds, minimal points and CR-manifolds with the microlocal analytic extension property.J. Geom. Anal., v.6, n. 4, 555–611, (1996).

    MathSciNet  MATH  Google Scholar 

  25. Lawrence, M. Hulls of tame Cantor sets. To appear.

  26. Lehner, J. Discontinuous groups and automorphic functions,Math. Surv.,8, AMS, Providence, RI, (1964).

  27. Lupacciolu, G. Characterization of removable sets in strongly pseudoconvex boundaries,Ark. för Math.,32, 455–473, (1994).

    Article  MathSciNet  MATH  Google Scholar 

  28. Lupacciolu, G. and Stout, E.M. Removable Singularities for\(\bar \partial _b \), Proc. Mittag-Leffler special year in complex analysis, Princeton University Press, Princeton, NJ, 1993.

    Google Scholar 

  29. Marshall, D. Removable sets for bounded analytic functions,Lecture Notes in Math.,1043, 485–490, Springer, Berlin, (1984).

    Google Scholar 

  30. Narasimhan, R. Introduction to the theory of analytic spaces,Lect. Notes in Math.,25, Springer, Berlin, (1966).

    Google Scholar 

  31. Painlevé, P. Sur les lignes singulières des fondons analytiques,Ann. Fac. Sci. Toulouse,2, (1888).

  32. Slodkowski, Z. An analytic set valued selection and its application to the corona theorem, to polynomial hulls and joint spectra,Trans. Am. Math. Soc.,294(1), 367–377, (1986).

    Article  MathSciNet  MATH  Google Scholar 

  33. Springer, G.Introduction to Riemann Surfaces, Addison-Wesley, Massachusetts, 1957.

    MATH  Google Scholar 

  34. Stolzenberg, G. Polynomially and rationally convex sets,Acta Math.,109, 259–289, (1963).

    Article  MathSciNet  MATH  Google Scholar 

  35. Stolzenberg, G. Uniform approximation on smooth curves,Acta Math.,115, 185–198, (1966).

    Article  MathSciNet  MATH  Google Scholar 

  36. Stout, E.L.Removable Singularities for the Boundary Values of Holomorphic Functions of Several Complex Variables, Proc. Mittag-Leffier special year in complex variables, Princeton University Press, Princeton, NJ, 1993.

    Google Scholar 

  37. Stout, E.L. Harmonic duality, hyperfunctions and removable singularities,Izvestiya Rus. Acad. Sci., Ser. Math.,59(6), 133–170, (1995).

    MathSciNet  Google Scholar 

  38. Sussmann, H.J. Orbits of families of vector fields and integrability of distributions,Trans. Am. Math. Soc.,180, 171–188, (1973).

    Article  MathSciNet  MATH  Google Scholar 

  39. Trépreau, J.-M. Sur la propagation des singularités dans les variétiés CR,Bull. Soc. Math. France,118, 403–450, (1990).

    MathSciNet  MATH  Google Scholar 

  40. Trépreau, J.-M.Holomorphic Extension of CR-Functions: A Survey, Progress in non-linear differential equations and their applications, The Danish-Swedish Analysis Seminar 1995. Birkhäuser, Boston, 1996.

    Google Scholar 

  41. Treves, F.Approximation and Representation of Functions and Distributions Annihilated by a System of Complex Vector Fields, École Polytech., Centre de Math., Palaiseau, France, 1981.

  42. Tumanov, A.E. Extension of CR-functions into a wedge from a manifold of finite type,Mat. Sbornik,136, 129–140, (1988).

    Google Scholar 

  43. Tumanov, A.E. Extension of CR-functions into a wedge,Mat. Sbornik,181, 951–964, (1990).

    MATH  Google Scholar 

  44. Tumanov, A.E. Connections and propagation of analyticity for CR-functions,Duke Math. J.,73, 1–24, (1994).

    Article  MathSciNet  MATH  Google Scholar 

  45. Vladimirov, V.S.Fonctions de Plusieurs Variables Complexes, Dunod, Paris, 1971.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut für Mathematik, Lehrstuhl Zahlentheorie, Humboldt-Universität zu Berlin, Jägerstraße 10/11, 10117, Berlin, Germany

    Burglind Jöricke

  2. Mathematical Department, Uppsala University, Box 480, SE-75106, Uppsala, Sweden

    Burglind Jöricke

Authors
  1. Burglind Jöricke
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Burglind Jöricke.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Jöricke, B. Boundaries of singularity sets, removable singularities, and CR-invariant subsets of CR-manifolds. J Geom Anal 9, 257–300 (1999). https://doi.org/10.1007/BF02921939

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02921939

Math Subject Classifications

Key Words and Phases

Search

Navigation