Advertisement

The Journal of Geometric Analysis

, Volume 28, Issue 2, pp 1052–1077 | Cite as

Local Criteria for Non-Embeddability of Levi-Flat Manifolds

  • Takayuki Koike
  • Noboru OgawaEmail author
Article

Abstract

We give local criteria for smooth non-embeddability of Levi-flat manifolds. For this purpose, we pose an analogue of Ueda theory on the neighborhood structure of hypersurfaces in complex manifolds with topologically trivial normal bundles.

Keywords

Levi-flat manifolds Ueda theory Pseudo-flat neighborhoods 

Mathematics Subject Classification

32V30 (Embeddings of CR manifolds) 37F75 (Holomorphic foliations and vector fields) 14B20 (Formal neighborhoods) 

Notes

Acknowledgements

The authors would like to thank the anonymous reviewers for their careful reading of our manuscript. The first author is supported by the Grant-in-Aid for Scientific Research (KAKENHI No.25-2869).

References

  1. 1.
    Adachi, M., Brinkschulte, J.: Curvature restrictions for Levi-flat real hypersurfaces in complex projective planes. Ann. Inst. Fourier (Grenoble) 65(6), 2547–2569 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Barrett, D.E.: Complex analytic realization of Reeb’s foliation of \(S^3\). Math. Z. 203, 355–361 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Barrett, D.E., Fornæss, J.-E.: On the smoothness of Levi-foliations. Publ. Mat. 32, 171–177 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Barrett, D.E., Inaba, T.: On the topology of compact smooth three-dimensional Levi-flat hypersurfaces. J. Geom. Anal. 2(6), 489–497 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Brunella, M.: On the dynamics of codimension one holomorphic foliations with ample normal bundle. Indiana Univ. Math. J. 57, 3101–3113 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bonatti, C., Langevin, R., Moussu, R.: Feuilletages de CP(n): de l’holonomie hyperbolique pour les minimaux exceptionnels. Inst. Hautes Études Sci. Publ. Math. 75, 123–134 (1992)CrossRefzbMATHGoogle Scholar
  7. 7.
    Cerveau, D.: Minimaux des feuilletages algébriques de \(\mathbb{C}\text{ P }^n\). Ann. Inst. Fourier 43, 1535–1543 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Candel, A., Conlon, L.: Foliations I and II. American Mathematical Society, Providence, RI (2003)CrossRefzbMATHGoogle Scholar
  9. 9.
    Claudon, B., Loray, F., Pereira, J.V., Touzet, F.: Compact leaves of codimension one holomorphic foliations on projective manifolds, arXiv:math/1512.06623
  10. 10.
    Camacho, C., Lins Neto, A., Sad, P.: Minimal sets of foliations on complex projective spaces. Inst. Hautes Études Sci. Publ. Math. No. 68, 187–203 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Della Sala, G.: Non-embeddability of certain classes of Levi flat manifolds. Osaka J. Math. 51, 161–169 (2014)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Deroin, B.: Hypersurfaces Levi-plates immergées dans les surfaces complexes de courbure positive. C. R. Math. Acad. Sci. Paris 337(12), 777–780 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Forstnerič, F., Laurent-Thiébaut, C.: Stein compacts in Levi-flat hypersurfaces. Trans. Am. Math. Sor. 360, 307–329 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Horiuchi, T., Mitsumatsu, Y.: Reeb components with complex leaves and their symmetries I : The automorphism groups and Schröder’s equation on the half line, arXiv:1605.08977
  15. 15.
    Kodaira, K., Spencer, D.C.: A theorem of completeness of characteristic systems of complete continuous systems. Am. J. Math. 81, 477–500 (1959)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Koike, T.: Toward a higher codimensional Ueda theory. Math. Z. 281(3), 967–991 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Lins-Neto, A.: A note on projective Levi flats and minimal sets of algebraic foliations. Ann. Inst. Fourier (Grenoble) 49, 1369–1385 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Meersseman, L., Verjovsky, A.: On the moduli space of certain smooth codimension-one foliations of the 5-sphere by complex surfaces. J. Reine Angew. Math. 632, 143–202 (2009)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Neeman, A.: Ueda theory: theorems and problems, vol. 81(415). Memoirs of the American Mathematical Society (1989)Google Scholar
  20. 20.
    Nemirovski, S.: Stein domains with Levi-flat boundaries in compact complex surfaces. Math. Notes 66, 522–525 (1999)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Ohsawa, T.: Levi flat hypersurfaces in complex manifolds, Complex analysis and digital geometry, Acta Univ. Upsaliensis Skr. Uppsala Univ. C Organ. Hist. vol. 86, pp. 223–231. Uppsala Universitet, Uppsala (2009)Google Scholar
  22. 22.
    Ohsawa, T.: A survey on Levi flat hypersurfaces. In: Complex Geometry and Dynamics, Abel Symposia, Vol. 10, pp. 211–225Google Scholar
  23. 23.
    Ohsawa, T.: Nonexistence of certain Levi flat hypersurfaces in Kähler manifolds from the viewpoint of positive normal bundles. Publ. RIMS Kyoto Univ. 49, 229–239 (2013)CrossRefzbMATHGoogle Scholar
  24. 24.
    Siegel, C.L.: Iterations of analytic functions. Ann. Math. 43, 607–612 (1942)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Siu, Y.: Nonexistence of smooth Levi-flat hypersurfaces in complex projective spaces of dimension \(\ge \) 3. Ann. Math. (2) 151(3), 1217–1243 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Ueda, T.: On the neighborhood of a compact complex curve with topologically trivial normal bundle. J. Math. Kyoto Univ. 22, 583–607 (1983)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Mathematica Josephina, Inc. 2017

Authors and Affiliations

  1. 1.Department of Mathematics, Graduate School of ScienceKyoto UniversityKyotoJapan
  2. 2.Department of MathematicsTokai UniversityHiratsuka-shiJapan

Personalised recommendations