Inventiones mathematicae

, Volume 203, Issue 1, pp 333–358 | Cite as

Rationally convex domains and singular Lagrangian surfaces in \(\mathbb {C}^2\)

Article

Abstract

We give a complete characterization of those disk bundles over surfaces which embed as rationally convex strictly pseudoconvex domains in \(\mathbb {C}^2\). We recall some classical obstructions and prove some deeper ones related to symplectic and contact topology. We explain the close connection to Lagrangian surfaces with isolated singularities and develop techniques for constructing such surfaces. Our proof also gives a complete characterization of Lagrangian surfaces with open Whitney umbrellas, answering a question first posed by Givental in 1986.

Notes

Acknowledgments

We would like to thank Yasha Eliashberg for suggesting this problem and for numerous informative discussions. We also thank Roger Casals and Emmy Murphy for enlightening conversations regarding Sect. 5.

References

  1. 1.
    Abouzaid, M, Seidel, P.: Altering symplectic manifolds by homologous recombination (2010). arXiv:1007.3281
  2. 2.
    Akbulut, S., Matveyev, R.: Exotic structures and adjunction inequality. Turkish J. Math. 21, 47–53 (1997)MathSciNetMATHGoogle Scholar
  3. 3.
    Audin, M.: Quelques remarques sur les surfaces lagrangiennes de Givental. J. Geom. Phys. 7, 583–598 (1990)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Castro, I., Lerma, A.M.: Hamiltonian stationary self-similar solutions for Lagrangian mean curvature flow in the complex Euclidean plane. Proc. Am. Math. Soc. 138, 1821–1832 (2010)Google Scholar
  5. 5.
    Chantraine, B.: Lagrangian concordance of Legendrian knots. Algebr. Geom. Topol. 10, 63–85 (2010)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Cieliebak, K., Eliashberg, Y.: From Stein to Weinstein and Back: Symplectic Geometry of Affine Complex Manifolds. American Mathematical Society, Providence, RI (2012)Google Scholar
  7. 7.
    Cieliebak, K., Eliashberg, Y.: The topology of rationally and polynomially convex domains. Invent. Math. 199, 215–238 (2015)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Duval, J., Sibony, N.: Polynomial convexity, rational convexity, and currents. Duke Math. J. 79, 487–513 (1995)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Ekholm, T.: Rational SFT, linearized legendrian contact homology, and lagrangian floer cohomology. In: Perspectives in Analysis, Geometry, and Topology. Progress in Mathematics, vol. 296, pp. 109–145. Springer, New York (2012)Google Scholar
  10. 10.
    Ekholm, T., Honda, K., Kálmán, T.: Legendrian knots and exact Lagrangian cobordisms (2012). arXiv:1212.1519
  11. 11.
    Eliashberg, Y.: Topological characterization of Stein manifolds of dimension > 2. Int. J. Math. 1, 29–46 (1990)Google Scholar
  12. 12.
    Eliashberg, Y., Fraser, M.: Topologically trivial Legendrian knots. J. Symplectic Geom. 7, 77–127 (2009)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Eliashberg, Y., Murphy, E.: Lagrangian caps. Geom. Funct. Anal. 23, 1483–1514 (2013)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Eliashberg, Y., Gromov, M.: Lagrangian intersection theory: finite-dimensional approach. Transl. Am. Math. Soc. Ser. 2(186), 27–118 (1998)MathSciNetGoogle Scholar
  15. 15.
    Etnyre, J.B.: Legendrian and Transversal Knots. Handbook of Knot Theory. pp. 105–185. Elsevier, Amsterdam (2005)Google Scholar
  16. 16.
    Forstnerič, F.: Complex tangents of real surfaces in complex surfaces. Duke Math. J. 67, 353–376 (1992)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Forstnerič, F.: Stein domains in complex surfaces. J. Geom. Anal. 13, 77–94 (2003)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Forstnerič, F.: Stein Manifolds and Holomorphic Mappings: The Homotopy Principle in Complex Analysis. In: Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, vol. 56. Springer, Berlin, New York (2011)Google Scholar
  19. 19.
    Ghiggini, P., Lisca, P., Stipsicz, A.: Tight contact structures on some small Seifert fibered 3-manifolds. Am. J. Math. 129, 1403–1447 (2007)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Giroux, E.: Une infinité de structures de contact tendues sur une infinité de variétés. Invent. Math. 135, 789–802 (1999)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Givental, A.: Lagrangian imbeddings of surfaces and unfolded Whitney umbrella. Funct. Anal. Appl. 20, 197–203 (1986)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Gromov, M.: Pseudo holomorphic curves in symplectic manifolds. Invent. Math. 82, 307–347 (1985)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Honda, K.: On the classification of tight contact structures II. J. Differ. Geom. 55, 83–143 (2000)MathSciNetMATHGoogle Scholar
  24. 24.
    Eliashberg, Y.: Unique holomorphically fillable contact structure on the 3-torus. Int. Math. Res. Notices 1996(2), 77–82 (1996)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Johns, J.: Morse-Bott handle attachments and plumbing (2009). http://www.cims.nyu.edu/~jjohns/MorseBottHandlesB.pdf
  26. 26.
    Kanda, Y.: The classification of tight contact structures on the 3-torus. Commun. Anal. Geom. 5, 413–438 (1997)MathSciNetMATHGoogle Scholar
  27. 27.
    Lai, H.F.: Characteristic classes of real manifolds immersed in complex manifolds. Trans. Am. Math. Soc. 172, 1–33 (1972)Google Scholar
  28. 28.
    Lin, F.: Exact Lagrangian caps of Legendrian knots (2013). arXiv:1309.5101
  29. 29.
    Lisca, P., Matić, G.: Tight contact structures and Seiberg–Witten invariants. Invent. Math. 129, 509–525 (1997)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Massey, W.: Proof of a conjecture of Whitney. Pac. J. Math. 31, 143–156 (1969)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    McDuff, D.: The structure of rational and ruled symplectic 4-manifolds. J. Am. Math. Soc. 3, 679–712 (1990)MathSciNetMATHGoogle Scholar
  32. 32.
    McDuff, D.: Symplectic manifolds with contact type boundaries. Invent. Math. 103, 651–671 (1991)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    McDuff, D., Salamon, D.: J-Holomorphic Curves and Symplectic Topology. American Mathematical Society, Providence, RI (2012)Google Scholar
  34. 34.
    Murphy, E.: Loose Legendrian embeddings in high dimensional contact manifolds (2012). arXiv:1201.2245
  35. 35.
    Nemirovski, S.: Complex analysis and differential topology on complex surfaces. Russ. Math. Surv. 54, 729–752 (1999)CrossRefGoogle Scholar
  36. 36.
    Nemirovski, S.: Adjunction inequality and coverings of Stein surfaces. Turk. J. Math. 27, 161–172 (2003)MathSciNetMATHGoogle Scholar
  37. 37.
    Nemirovski, S.: Finite unions of balls in \({\mathbb{C}}^{n}\) are rationally convex. Russ. Math. Surv. 63, 381–382 (2008)CrossRefMATHGoogle Scholar
  38. 38.
    Nemirovski, S.: Lagrangian Klein bottles in \({\mathbb{R}}^{2n}\). Geom. Funct. Anal. 19, 902–909 (2009)MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Neumann, W.D.: A calculus for plumbing applied to the topology of complex surface singularities and degenerating complex curves. Trans. Am. Math. Soc. 268, 299–344 (1981)CrossRefMATHGoogle Scholar
  40. 40.
    Ozbagci, B., András, B.S.: Surgery on Contact 3-Manifolds and Stein Surfaces. Springer, New York (2004)CrossRefMATHGoogle Scholar
  41. 41.
    Shevchishin, V.V.: Lagrangian embeddings of the Klein bottle and combinatorial properties of mapping class groups. Izvestiya: Mathematics 73, 797–859 (2009)MathSciNetCrossRefMATHGoogle Scholar
  42. 42.
    Stout, E.L.: Polynomial Convexity. Birkhäuser, Boston (2007)MATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Steklov Mathematical InstituteMoscowRussia
  2. 2.Fakultät für MathematikRuhr-Universität BochumBochumGermany
  3. 3.Department of MathematicsStanford UniversityStanfordUSA

Personalised recommendations