Skip to main content
SpringerLink
Log in

Tight Beltrami fields with symmetry

  • Original paper
  • Published:
Geometriae Dedicata Aims and scope Submit manuscript

Abstract

Let M be a compact orientable Seifered fibered 3-manifold without a boundary, and α an S 1-invariant contact form on M. In a suitable adapted Riemannian metric to α, we provide a bound for the volume Vol(M) and the curvature, which implies the universal tightness of the contact structure ξ = ker α.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aronszajn, N.: A unique continuation theorem for solutions of elliptic partial differential equations or inequalities of second order. J. Math. Pures Appl. (9) 36, 235–249 (1957)

    MathSciNet  MATH  Google Scholar 

  2. Bär, C.: On nodal sets for Dirac and Laplace operators. Comm. Math. Phys. 188(3), 709–721 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  3. Belgun, F.: Normal CR structures on S 3. Math. Z. 244(1), 125–151 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  4. Blair, D.: Riemannian geometry of contact and symplectic manifolds, vol. 203 of Progress in Mathematics. Birkhäuser Boston Inc., Boston, MA (2002)

    Google Scholar 

  5. Brin, M. Seifert Fibered Spaces: Notes for a Course given in the Spring of 1993. Online notes

  6. Bryant, R.L., Chern, S.S., Gardner, R.B., Goldschmidt, H.L., Griffiths, P.A.: Exterior differential systems, vol. 18 of Mathematical Sciences Research Institute Publications. Springer-Verlag, New York (1991)

    Google Scholar 

  7. Buser, P.: Riemannsche Flächen mit Eigenwerten in (0, 1/4). Comment. Math. Helv. 52(1), 25–34 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chavel, I.: Eigenvalues in Riemannian Geometry, vol. 115 of Pure and Applied Mathematics. Academic Press Inc., Orlando, FL (1984). Including a chapter by Burton Randol, With an Appendix by Jozef Dodziuk

  9. Chern, S.S., Hamilton, R.S.: On Riemannian metrics adapted to three-dimensional contact manifolds. In: Workshop Bonn 1984 (Bonn, 1984), vol. 1111 of Lecture Notes in Math., pp. 279–308. Springer, Berlin (1985)

  10. Dong, R.T.: Nodal sets of eigenfunctions on Riemann surfaces. J. Differential Geom. 36(2), 493–506 (1992)

    MathSciNet  MATH  Google Scholar 

  11. Eliashberg, Y.: Classification of overtwisted contact structures on 3-manifolds. Invent. Math. 98(3), 623–637 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  12. Epstein, D.B.A.: Periodic flows on three-manifolds. Ann. Math. (2) 95, 66–82 (1972)

    Article  Google Scholar 

  13. Etnyre, J., Ghrist, R.: Contact topology and hydrodynamics. I. Beltrami fields and the Seifert conjecture. Nonlinearity 13(2), 441–458 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  14. Etnyre, J., Ghrist, R.: Contact topology and hydrodynamics. III. Knotted orbits. Trans. Amer. Math. Soc. 352(12), 5781–5794 (2000) (electronic)

    Article  MathSciNet  MATH  Google Scholar 

  15. Etnyre, J., Ghrist, R.: Contact topology and hydrodynamics. II. Solid tori. Ergodic Theory Dyn. Syst. 22(3), 819–833 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  16. Etnyre, J., Honda, K.: On the nonexistence of tight contact structures. Ann. Math. (2) 153(3), 749–766 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  17. Evans, L.C.: Partial Differential Equations, vol. 19 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI (1998)

    Google Scholar 

  18. Geiges, H.: Normal contact structures on 3-manifolds. Tohoku Math. J. (2) 49(3), 415–422 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  19. Geiges, H.: Contact geometry. In: Handbook of Differential Geometry, vol. II, pp. 315–382. Elsevier/North-Holland, Amsterdam (2006)

  20. Ghrist, R., Komendarczyk, R.: Overtwisted energy-minimizing curl eigenfields. Nonlinearity 19(1), 41–51 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  21. Gilbarg, D., Trudinger, N.: Elliptic partial differential equations of second order, vol. 224 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 2nd edn. Springer-Verlag, Berlin (1983)

    Google Scholar 

  22. Gilkey, P., Leahy, J., Park, J.: Spinors, spectral geometry, and Riemannian submersions, vol. 40 of Lecture Notes Series. Seoul National University Research Institute of Mathematics Global Analysis Research Center, Seoul (1998)

    Google Scholar 

  23. Giroux, E.: Convexité en topologie de contact. Comment. Math. Helv. 66(4), 637–677 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  24. Giroux, E.: Structures de contact sur les variétés fibrées en cercles audessus d’une surface. Comment. Math. Helv. 76(2), 218–262 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  25. Honda, K.: On the classification of tight contact structures. I. Geom. Topol. 4, 309–368 (2000) (electronic)

    Article  MathSciNet  MATH  Google Scholar 

  26. Honda, K.: On the classification of tight contact structures. II. J. Differential Geom. 55(1), 83–143 (2000)

    MathSciNet  MATH  Google Scholar 

  27. Jost, J.: Riemannian Geometry and Geometric Analysis, 3rd edn. Universitext. Springer-Verlag, Berlin (2002)

    Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  29. Komendarczyk, R.: Nodal sets and contact structures. Ph.D. thesis Georgia Tech’06: http://smartech.gatech.edu/handle/1853/11517, the extended version to appear as a VDM monograph (2006)

  30. Komendarczyk, R.: On the contact geometry of nodal sets. Trans. Amer. Math. Soc. 358(6), 2399–2413 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  31. McDuff, D., Salamon, D.: Introduction to Symplectic Topology, 2nd edn. Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, New York (1998)

  32. Nicolaescu, L.: Adiabatic limits of the Seiberg-Witten equations on Seifert manifolds. Comm. Anal. Geom. 6(2), 331–392 (1998)

    MathSciNet  MATH  Google Scholar 

  33. Savo, A.: Eigenvalue estimates and nodal length of eigenfunctions. In: Steps in Differential Geometry (Debrecen, 2000), pp. 295–301. Inst. Math. Inform., Debrecen (2001)

  34. Scott, P.: The geometries of 3-manifolds. Bull. London Math. Soc. 15(5), 401–487 (1983)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Pennsylvania, Philadelphia, PA, USA

    Rafal Komendarczyk

Authors
  1. Rafal Komendarczyk
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Rafal Komendarczyk.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Komendarczyk, R. Tight Beltrami fields with symmetry. Geom Dedicata 134, 217–238 (2008). https://doi.org/10.1007/s10711-008-9258-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10711-008-9258-9

Keywords

Mathematics Subject Classification (2000)

Search

Navigation