Mathematische Annalen

, Volume 355, Issue 3, pp 1177–1199 | Cite as

Singular set of a Levi-flat hypersurface is Levi-flat



We study the singular set of a singular Levi-flat real-analytic hypersurface. We prove that the singular set of such a hypersurface is Levi-flat in the appropriate sense. We also show that if the singular set is small enough, then the Levi-foliation extends to a singular codimension one holomorphic foliation of a neighborhood of the hypersurface.

Mathematics Subject Classification (2000)

32C07 32V40 37F75 32B20 14P15 


  1. 1.
    Baouendi, M.S., Ebenfelt, P., Rothschild, L.P.: CR automorphisms of real analytic manifolds in complex space. Commun. Anal. Geom. 6(2), 291–315 (1998)MathSciNetMATHGoogle Scholar
  2. 2.
    Baouendi, M.S., Ebenfelt, P., Rothschild, L.P.: Real submanifolds in complex space and their mappings. In: Princeton Mathematical Series, vol. 47. Princeton University Press, Princeton (1999)Google Scholar
  3. 3.
    Bedford, E.: Holomorphic continuation of smooth functions over Levi-flat hypersurfaces. Trans. Am. Math. Soc. 232, 323–341 (1977)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Bierstone, E., Milman, P.D.: Semianalytic and subanalytic sets. Inst. Hautes Études Sci. Publ. Math. 67, 5–42 (1988)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Brunella, M.: Singular Levi-flat hypersurfaces and codimension one foliations. Ann. Sci. Norm. Super. Pisa Cl. Sci. (5) 6(4), 661–672 (2007)MathSciNetMATHGoogle Scholar
  6. 6.
    Burns, D., Gong, X.: Singular Levi-flat real analytic hypersurfaces. Am. J. Math. 121(1), 23–53 (1999)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Camacho, C., Neto, A.L.: Geometric Theory of Foliations. Birkhäuser, Boston (1985). Translated from the Portuguese by Sue E. GoodmanMATHGoogle Scholar
  8. 8.
    Cerveau, D., Neto, A.L.: Local Levi-flat hypersurfaces invariants by a codimension one holomorphic foliation. Am. J. Math. 133(3), 677–716 (2011)MATHCrossRefGoogle Scholar
  9. 9.
    D’Angelo, J.P.: Several complex variables and the geometry of real hypersurfaces. In: Studies in Advanced Mathematics. CRC Press, Boca Raton (1993)Google Scholar
  10. 10.
    Diederich, K., Fornæss, J.E.: Pseudoconvex domains with real-analytic boundary. Ann. Math. (2) 102(2), 384–371 (1978)Google Scholar
  11. 11.
    Fernández-Pérez, A.: On normal forms of singular Levi-flat real analytic hypersurfaces. Bull. Braz. Math. Soc. (N.S.) 42(1), 75–85 (2011)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Jarnicki, M., Pflug, P.: Extension of holomorphic functions. In: de Gruyter Expositions in Mathematics, vol. 34. Walter de Gruyter& Co., Berlin (2000)CrossRefGoogle Scholar
  13. 13.
    Kohn, J.J.: Subellipticity of the \(\bar{\partial }\)-Neumann problem on pseudo-convex domains: sufficient conditions. Acta Math. 142(1–2), 79–122 (1979)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Krantz, S.G.: Function theory of several complex variables. AMS Chelsea Publishing, Providence (2001) (Reprint of the 1992 edition)Google Scholar
  15. 15.
    Lebl, J.: Nowhere minimal CR submanifolds and Levi-flat hypersurfaces. J. Geom. Anal. 17(2), 321–342 (2007)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Lebl, J.: Singularities and Complexity in CR Geometry, Ph.D. thesis, University of California at San Diego (2007)Google Scholar
  17. 17.
    Lebl, J.: Algebraic Levi-flat hypervarieties in complex projective space. J. Geom. Anal. 22, 410–432 (2012)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Neto, A.L.: A note on projective Levi flats and minimal sets of algebraic foliations. Ann. Inst. Fourier (Grenoble) 49(4), 1369–1385 (1999)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Siu, Y.-T.: Techniques of extension of analytic objects. In: Lecture Notes in Pure and Applied Mathematics, vol. 8. Marcel Dekker Inc., New York (1974)Google Scholar
  20. 20.
    Hassler, W.: Complex analytic varieties. Addison-Wesley Publishing Co., Reading (1972)MATHGoogle Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of California at San DiegoLa JollaUSA
  2. 2.Department of MathematicsUniversity of WisconsinMadisonUSA

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