Mathematische Annalen

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

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

  • Jiří Lebl


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 



The author would like to acknowledge Peter Ebenfelt for suggesting the study of this problem when the author was still a graduate student, and also for many conversations on the topic. The author would also like to thank Xianghong Gong, John P. D’Angelo, Salah Baouendi, Linda Rothschild, and Arturo Fernández-Pérez for fruitful discussions on topics related to this research and suggestions on the manuscript.


  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)MathSciNetzbMATHGoogle 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)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Bierstone, E., Milman, P.D.: Semianalytic and subanalytic sets. Inst. Hautes Études Sci. Publ. Math. 67, 5–42 (1988)MathSciNetzbMATHCrossRefGoogle 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)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Burns, D., Gong, X.: Singular Levi-flat real analytic hypersurfaces. Am. J. Math. 121(1), 23–53 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Camacho, C., Neto, A.L.: Geometric Theory of Foliations. Birkhäuser, Boston (1985). Translated from the Portuguese by Sue E. GoodmanzbMATHGoogle 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)zbMATHCrossRefGoogle 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)MathSciNetzbMATHCrossRefGoogle 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)MathSciNetzbMATHCrossRefGoogle 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)MathSciNetzbMATHCrossRefGoogle 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)MathSciNetzbMATHCrossRefGoogle 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)MathSciNetzbMATHCrossRefGoogle 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)zbMATHGoogle 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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