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

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Abstract

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.

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References

  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)

    MathSciNet  MATH  Google Scholar 

  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. Bedford, E.: Holomorphic continuation of smooth functions over Levi-flat hypersurfaces. Trans. Am. Math. Soc. 232, 323–341 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bierstone, E., Milman, P.D.: Semianalytic and subanalytic sets. Inst. Hautes Études Sci. Publ. Math. 67, 5–42 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  5. Brunella, M.: Singular Levi-flat hypersurfaces and codimension one foliations. Ann. Sci. Norm. Super. Pisa Cl. Sci. (5) 6(4), 661–672 (2007)

    MathSciNet  MATH  Google Scholar 

  6. Burns, D., Gong, X.: Singular Levi-flat real analytic hypersurfaces. Am. J. Math. 121(1), 23–53 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  7. Camacho, C., Neto, A.L.: Geometric Theory of Foliations. Birkhäuser, Boston (1985). Translated from the Portuguese by Sue E. Goodman

    MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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. Diederich, K., Fornæss, J.E.: Pseudoconvex domains with real-analytic boundary. Ann. Math. (2) 102(2), 384–371 (1978)

    Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  12. Jarnicki, M., Pflug, P.: Extension of holomorphic functions. In: de Gruyter Expositions in Mathematics, vol. 34. Walter de Gruyter& Co., Berlin (2000)

    Book  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  14. Krantz, S.G.: Function theory of several complex variables. AMS Chelsea Publishing, Providence (2001) (Reprint of the 1992 edition)

  15. Lebl, J.: Nowhere minimal CR submanifolds and Levi-flat hypersurfaces. J. Geom. Anal. 17(2), 321–342 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  16. Lebl, J.: Singularities and Complexity in CR Geometry, Ph.D. thesis, University of California at San Diego (2007)

  17. Lebl, J.: Algebraic Levi-flat hypervarieties in complex projective space. J. Geom. Anal. 22, 410–432 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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. Hassler, W.: Complex analytic varieties. Addison-Wesley Publishing Co., Reading (1972)

    MATH  Google Scholar 

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Acknowledgments

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.

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  1. Jiří Lebl

    Present address: Department of Mathematics, University of Wisconsin, Madison, WI, 53706, USA

Authors and Affiliations

  1. Department of Mathematics, University of California at San Diego, La Jolla, CA, 92093-0112, USA

    Jiří Lebl

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  1. Jiří Lebl
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Correspondence to Jiří Lebl.

Additional information

J. Lebl was in part supported by NSF grant DMS 0900885.

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Lebl, J. Singular set of a Levi-flat hypersurface is Levi-flat. Math. Ann. 355, 1177–1199 (2013). https://doi.org/10.1007/s00208-012-0821-1

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  • DOI: https://doi.org/10.1007/s00208-012-0821-1

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