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

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.

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Notes

  1. In the terminology of Cieliebak–Eliashberg, the closure of a smoothly bounded strictly pseudoconvex domain is called an \(i\) -convex domain, and a strictly plurisubharmonic function is called an i-convex function. In this paper, “domain” means open connected set.

  2. We reserve the term knot for links with a single connected component.

  3. The sign discrepancy arises because, for an oriented Lagrangian plane \(P \subset \mathbb {C}^2\), we have \(P \oplus iP = \mathbb {C}^2\), but the induced orientation on \(P \oplus iP\) as a direct sum is the opposite of the orientation on \(\mathbb {C}^2\) as a complex vector space.

References

  1. Abouzaid, M, Seidel, P.: Altering symplectic manifolds by homologous recombination (2010). arXiv:1007.3281

  2. Akbulut, S., Matveyev, R.: Exotic structures and adjunction inequality. Turkish J. Math. 21, 47–53 (1997)

    MathSciNet  MATH  Google Scholar 

  3. Audin, M.: Quelques remarques sur les surfaces lagrangiennes de Givental. J. Geom. Phys. 7, 583–598 (1990)

    MathSciNet  Article  MATH  Google Scholar 

  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)

  5. Chantraine, B.: Lagrangian concordance of Legendrian knots. Algebr. Geom. Topol. 10, 63–85 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  6. Cieliebak, K., Eliashberg, Y.: From Stein to Weinstein and Back: Symplectic Geometry of Affine Complex Manifolds. American Mathematical Society, Providence, RI (2012)

  7. Cieliebak, K., Eliashberg, Y.: The topology of rationally and polynomially convex domains. Invent. Math. 199, 215–238 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  8. Duval, J., Sibony, N.: Polynomial convexity, rational convexity, and currents. Duke Math. J. 79, 487–513 (1995)

    MathSciNet  Article  MATH  Google Scholar 

  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)

  10. Ekholm, T., Honda, K., Kálmán, T.: Legendrian knots and exact Lagrangian cobordisms (2012). arXiv:1212.1519

  11. Eliashberg, Y.: Topological characterization of Stein manifolds of dimension > 2. Int. J. Math. 1, 29–46 (1990)

  12. Eliashberg, Y., Fraser, M.: Topologically trivial Legendrian knots. J. Symplectic Geom. 7, 77–127 (2009)

    MathSciNet  Article  MATH  Google Scholar 

  13. Eliashberg, Y., Murphy, E.: Lagrangian caps. Geom. Funct. Anal. 23, 1483–1514 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  14. Eliashberg, Y., Gromov, M.: Lagrangian intersection theory: finite-dimensional approach. Transl. Am. Math. Soc. Ser. 2(186), 27–118 (1998)

    MathSciNet  Google Scholar 

  15. Etnyre, J.B.: Legendrian and Transversal Knots. Handbook of Knot Theory. pp. 105–185. Elsevier, Amsterdam (2005)

  16. Forstnerič, F.: Complex tangents of real surfaces in complex surfaces. Duke Math. J. 67, 353–376 (1992)

    MathSciNet  Article  MATH  Google Scholar 

  17. Forstnerič, F.: Stein domains in complex surfaces. J. Geom. Anal. 13, 77–94 (2003)

    MathSciNet  Article  MATH  Google Scholar 

  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)

  19. Ghiggini, P., Lisca, P., Stipsicz, A.: Tight contact structures on some small Seifert fibered 3-manifolds. Am. J. Math. 129, 1403–1447 (2007)

    MathSciNet  Article  MATH  Google Scholar 

  20. Giroux, E.: Une infinité de structures de contact tendues sur une infinité de variétés. Invent. Math. 135, 789–802 (1999)

    MathSciNet  Article  MATH  Google Scholar 

  21. Givental, A.: Lagrangian imbeddings of surfaces and unfolded Whitney umbrella. Funct. Anal. Appl. 20, 197–203 (1986)

    MathSciNet  Article  MATH  Google Scholar 

  22. Gromov, M.: Pseudo holomorphic curves in symplectic manifolds. Invent. Math. 82, 307–347 (1985)

    MathSciNet  Article  MATH  Google Scholar 

  23. Honda, K.: On the classification of tight contact structures II. J. Differ. Geom. 55, 83–143 (2000)

    MathSciNet  MATH  Google Scholar 

  24. Eliashberg, Y.: Unique holomorphically fillable contact structure on the 3-torus. Int. Math. Res. Notices 1996(2), 77–82 (1996)

    MathSciNet  Article  MATH  Google Scholar 

  25. Johns, J.: Morse-Bott handle attachments and plumbing (2009). http://www.cims.nyu.edu/~jjohns/MorseBottHandlesB.pdf

  26. Kanda, Y.: The classification of tight contact structures on the 3-torus. Commun. Anal. Geom. 5, 413–438 (1997)

    MathSciNet  MATH  Google Scholar 

  27. Lai, H.F.: Characteristic classes of real manifolds immersed in complex manifolds. Trans. Am. Math. Soc. 172, 1–33 (1972)

  28. Lin, F.: Exact Lagrangian caps of Legendrian knots (2013). arXiv:1309.5101

  29. Lisca, P., Matić, G.: Tight contact structures and Seiberg–Witten invariants. Invent. Math. 129, 509–525 (1997)

    MathSciNet  Article  MATH  Google Scholar 

  30. Massey, W.: Proof of a conjecture of Whitney. Pac. J. Math. 31, 143–156 (1969)

    MathSciNet  Article  MATH  Google Scholar 

  31. McDuff, D.: The structure of rational and ruled symplectic 4-manifolds. J. Am. Math. Soc. 3, 679–712 (1990)

    MathSciNet  MATH  Google Scholar 

  32. McDuff, D.: Symplectic manifolds with contact type boundaries. Invent. Math. 103, 651–671 (1991)

    MathSciNet  Article  MATH  Google Scholar 

  33. McDuff, D., Salamon, D.: J-Holomorphic Curves and Symplectic Topology. American Mathematical Society, Providence, RI (2012)

  34. Murphy, E.: Loose Legendrian embeddings in high dimensional contact manifolds (2012). arXiv:1201.2245

  35. Nemirovski, S.: Complex analysis and differential topology on complex surfaces. Russ. Math. Surv. 54, 729–752 (1999)

    Article  Google Scholar 

  36. Nemirovski, S.: Adjunction inequality and coverings of Stein surfaces. Turk. J. Math. 27, 161–172 (2003)

    MathSciNet  MATH  Google Scholar 

  37. Nemirovski, S.: Finite unions of balls in \({\mathbb{C}}^{n}\) are rationally convex. Russ. Math. Surv. 63, 381–382 (2008)

    Article  MATH  Google Scholar 

  38. Nemirovski, S.: Lagrangian Klein bottles in \({\mathbb{R}}^{2n}\). Geom. Funct. Anal. 19, 902–909 (2009)

    MathSciNet  Article  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  40. Ozbagci, B., András, B.S.: Surgery on Contact 3-Manifolds and Stein Surfaces. Springer, New York (2004)

    Book  MATH  Google Scholar 

  41. Shevchishin, V.V.: Lagrangian embeddings of the Klein bottle and combinatorial properties of mapping class groups. Izvestiya: Mathematics 73, 797–859 (2009)

    MathSciNet  Article  MATH  Google Scholar 

  42. Stout, E.L.: Polynomial Convexity. Birkhäuser, Boston (2007)

    MATH  Google Scholar 

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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.

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Correspondence to Kyler Siegel.

Additional information

The first author was partially supported by DFG project SFB/TR-12 and RFBR Grant 14-01-00709-a. The second author was partially supported by NSF Grant DGE-114747.

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Nemirovski, S., Siegel, K. Rationally convex domains and singular Lagrangian surfaces in \(\mathbb {C}^2\) . Invent. math. 203, 333–358 (2016). https://doi.org/10.1007/s00222-015-0598-4

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Keywords

  • Contact Structure
  • Pseudoconvex Domain
  • Klein Bottle
  • Lagrangian Surface
  • Lagrangian Torus