Archiv der Mathematik

, Volume 113, Issue 6, pp 661–670 | Cite as

Stein and Weinstein structures on disk cotangent bundles of surfaces

  • Burak OzbagciEmail author


Gompf (Ann Math (2) 148(2):619–693, 1998) describes a Stein domain structure on the disk cotangent bundle of any closed surface S by a Legendrian handlebody diagram. We prove that Gompf’s Stein domain is symplectomorphic to the disk cotangent bundle equipped with its canonical symplectic structure and the boundary of this domain is contactomorphic to the unit cotangent bundle of S equipped with its canonical contact structure. As a corollary, we obtain a surgery diagram for the canonical contact structure on the unit cotangent bundle of S.


Stein structure Contact structure Cotangent bundle Weinstein structure Surgery diagram 

Mathematics Subject Classification




We would like to thank R.E. Gompf and A.I. Stipsicz for helpful comments on a draft of this paper.


  1. 1.
    Cieliebak, K., Eliashberg, Y.: From Stein to Weinstein and Back. Symplectic Geometry of Affine Complex Manifolds. American Mathematical Society Colloquium Publications, vol. 59. American Mathematical Society, Providence (2012)zbMATHGoogle Scholar
  2. 2.
    Ding, F., Geiges, H.: Handle moves in contact surgery diagrams. J. Topol. 2(1), 105–122 (2009)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Eliashberg, Y.: Unique holomorphically fillable contact structure on the \(3\)-torus. Int. Math. Res. Not. 2, 77–82 (1996)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Giroux, E.: Structures de contact sur les variétés fibrées en cercles audessus d’une surface. (French) [Contact structures on manifolds that are circle-bundles over a surface]. Comment. Math. Helv. 76(2), 218–262 (2001)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Gompf, R.E.: Handlebody construction of Stein surfaces. Ann. Math. (2) 148(2), 619–693 (1998)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Gompf, R.E., Stipsicz, A.I.: \(4\)-Manifolds and Kirby Calculus. Graduate Studies in Mathematics, vol. 20. American Mathematical Society, Providence (1999)zbMATHGoogle Scholar
  7. 7.
    Hind, R.: Holomorphic filling of \(\mathbb{R}P^3\). Commun. Contemp. Math. 2(3), 349–363 (2000)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Honda, K.: On the classification of tight contact structures. I. Geom. Topol. 4, 309–368 (2000)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Honda, K.: On the classification of tight contact structures. II. J. Differ. Geom. 55(1), 83–143 (2000)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Li, T.-J., Mak, C.Y., Yasui, K.: Calabi–Yau caps, uniruled caps and symplectic fillings. Proc. Lond. Math. Soc. (3) 114(1), 159–187 (2017)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Lisca, P., Stipsicz, A.I.: Tight, not semi-fillable contact circle bundles. Math. Ann. 328(1–2), 285–298 (2004)MathSciNetCrossRefGoogle Scholar
  12. 12.
    McDuff, D.: The structure of rational and ruled symplectic \(4\)-manifolds. J. Am. Math. Soc. 3(3), 679–712 (1990)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Ozbagci, B., Stipsicz, A.I.: Surgery on Contact 3-Manifolds and Stein Surfaces. Bolyai Society Mathematical Studies, vol. 13. Springer, Berlin; János Bolyai Mathematical Society, Budapest (2004)Google Scholar
  14. 14.
    Plamenevskaya, O., Van Horn-Morris, J.: Planar open books, monodromy factorizations and symplectic fillings. Geom. Topol. 14(4), 2077–2101 (2010)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Wendl, C.: Strongly fillable contact manifolds and J-holomorphic foliations. Duke Math. J. 151(3), 337–384 (2010)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Wendl, C.: Holomorphic Curves in Low Dimensions: From Symplectic Ruled Surfaces to Planar Contact Manifolds. Lecture Notes in Mathematics. Springer, New York (2018)CrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Department of MathematicsKoç UniversityIstanbulTurkey

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