Skip to main content
SpringerLink
Log in

Perfect Reeb flows and action–index relations

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

Abstract

We study non-degenerate Reeb flows arising from perfect contact forms, i.e., the forms with vanishing contact homology differential. In particular, we obtain upper bounds on the number of simple closed Reeb orbits for such forms on a variety of contact manifolds and certain action–index resonance relations for the standard contact sphere. Using these results, we reprove a theorem due to Bourgeois, Cieliebak and Ekholm characterizing perfect Reeb flows on the standard contact three-sphere as non-degenerate Reeb flows with exactly two simple closed orbits.

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. Abreu, M., Macarini, L.: Contact homology of good toric contact manifolds. Compos. Math. 148, 304–334 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bourgeois, F.: Morse-Bott Approach to Contact Homology, Ph.D. thesis, Stanford (2002)

  3. Bourgeois, F.: A survey of contact homology. CRM Proc. Lect. Notes 49, 45–60 (2009)

    MathSciNet  Google Scholar 

  4. Bourgeois, F., Cieliebak, K., Ekholm, T.: A note on Reeb dynamics on the tight 3-sphere. J. Mod. Dyn. 1, 597–613 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bourgeois, F., Eliashberg, Y., Hofer, H., Wysocki, K., Zehnder, E.: Compactness results in symplectic field theory. Geom. Topol. 7, 799–888 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  6. Bourgeois, F., Oancea, A.: An exact sequence for contact- and symplectic homology. Invent. Math. 175, 611–680 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  7. Bourgeois, F., Oancea, A.: \(S^1\)-equivariant symplectic homology and linearized contact homology, Preprint (2012). arXiv:1212.3731.

  8. Bramham, B., Hofer, H.: First steps towards a symplectic dynamics. Surv. Diff. Geom. 17, 127–178 (2012)

    Article  MathSciNet  Google Scholar 

  9. Cassels, J.W.S.: An Introduction to Diophantine Approximations, vol. 45. Cambridge University Press, Cambridge (1957)

    Google Scholar 

  10. Chance, M., Ginzburg, V.L., Gürel, B.Z.: Action-index relations for perfect Hamiltonian diffeomorphisms. J. Symplectic Geom. 11, 449–474 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  11. Cristofaro-Gardiner, D., Hutchings, M.: From one Reeb orbit to two, Preprint 2012, arXiv:1202.4839; to appear in J. Diff. Geom.

  12. Collier, B., Kerman, E., Reiniger, B.M., Turmunkh, B., Zimmer, A.: A symplectic proof of a theorem of Franks. Compos. Math. 148, 1069–1984 (2012)

    MathSciNet  Google Scholar 

  13. Ekeland, I., Hofer, H.: Convex Hamiltonian energy surfaces and their periodic trajectories. Commun. Math. Phys. 113, 419–469 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  14. Eliashberg, Y., Givental, A., Hofer, H.: Introduction to symplectic field theory. Geom. Funct. Anal., Special Volume. Part II, 560–673 (2000)

  15. Espina, J.: On the mean Euler characteristic of contact manifolds, Preprint (2010), arXiv:1011.4364; to appear in Int. J. Math.

  16. Foulon, P., Hasselblatt, B.: Contact Anosov flows on hyperbolic 3-manifolds. Geom. Topol. 17, 1225–1252 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  17. Franks, J.: Geodesics on \(S^2\) and periodic points of annulus homeomorphisms. Invent. Math. 108, 403–418 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  18. Franks, J.: Area preserving homeomorphisms of open surfaces of genus zero. N. Y. J. Math. 2, 1–19 (1996)

  19. Franks, J., Handel, M.: Periodic points of Hamiltonian surface diffeomorphisms. Geom. Topol. 7, 713–756 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  20. Gauntlett, J., Martelli, D., Sparks, J., Waldram, D.: A new infinite class of Sasaki–Einstein manifolds. Adv. Theor. Math. Phys. 8, 987–1000 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  21. Ginzburg, V.L., Gören, Y.: Iterated index and the mean Euler characteristic, Preprint (2013), arXiv:1311.0547

  22. Ginzburg, V.L., Gürel, B.Z.: Action and index spectra and periodic orbits in Hamiltonian dynamics. Geom. Topol. 13, 2745–2805 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  23. Ginzburg, V.L., Gürel, B.Z.: On the generic existence of periodic orbits in Hamiltonian dynamics. J. Mod. Dyn. 3, 595–610 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  24. Ginzburg, V.L., Gürel, B.Z.: Conley conjecture for negative monotone symplectic manifolds. Int. Math. Res. Not. IMRN (2011). doi:10.1093/imrn/rnr081

    Google Scholar 

  25. Ginzburg, V.L., Gürel, B.Z.: Hyperbolic fixed points and periodic orbits of Hamiltonian diffeomorphisms, Preprint (2012), arXiv:1208.1733; to appear in Duke Math. J.

  26. Ginzburg, V.L., Hein, D., Hryniewicz, U.L., Macarini, L.: Closed Reeb orbits on the sphere and symplectically degenerate maxima. Acta Math. Vietnam. 38, 55–78 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  27. Ginzburg, V.L., Kerman, E.: Homological resonances for Hamiltonian diffeomorphisms and Reeb flows. Int. Math. Res. Not. IMRN (2009). doi:10.1093/imrn/rnp120

  28. Guillemin, V., Ginzburg, V., Karshon, Y.: Moment Maps, Cobordisms, and Hamiltonian Group Actions, Mathematical Surveys and Monographs, vol. 98. American Mathematical Society, Providence, RI (2002)

    Book  Google Scholar 

  29. Gürel, B.Z.: Periodic orbits of Hamiltonian systems linear and hyperbolic at infinity, Preprint (2012), arXiv:1209.3529; to appear in Pacific J. Math.

  30. Gürel, B.Z.: On non-contractible periodic orbits of Hamiltonian diffeomorphisms. Bull. Lond. Math. Soc., 1227–1234 (2009). doi:10.1112/blms/bdt051

  31. Hardy, G.H., Wright, E.M.: An Introduction to the Theory of Numbers. Oxford Science Publications, Oxford (1979)

    MATH  Google Scholar 

  32. Hingston, N., Rademacher, H.-B.: Resonance for loop homology of spheres. J. Differ. Geom. 93, 133–174 (2013)

    MATH  MathSciNet  Google Scholar 

  33. Hofer, H.: Pseudoholomorphic curves in symplectizations with applications to the Weinstein conjecture in dimension three. Invent. Math. 114, 515–563 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  34. Hofer, H., Wysocki, K., Zehnder, E.: Unknotted periodic orbits for Reeb flows on the three-sphere. Topol. Methods Nonlinear Anal. 7, 219–244 (1996)

    MATH  MathSciNet  Google Scholar 

  35. Hofer, H., Wysocki, K., Zehnder, E.: The dynamics on the three dimensional strictly convex energy surfaces. Ann. Math. 148, 197–289 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  36. Hofer, H., Wysocki, K., Zehnder, E.: Finite energy foliations of tight three-spheres and Hamiltonian dynamics. Ann. Math. 157, 125–255 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  37. Hofer, H., Wysocki, K., Zehnder, E.: SC-smoothness, retractions and new models for smooth spaces. Discret. Contin. Dyn. Syst. 28, 665–788 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  38. Hofer, H., Wysocki, K., Zehnder, E.: Applications of polyfold theory I: the polyfolds of Gromov–Witten theory, Preprint (2011). arXiv:1107.2097

  39. Hofer, H., Zehnder, E.: Symplectic Invariants and Hamiltonian Dynamics. Birkäuser, Basel (1994)

    Book  MATH  Google Scholar 

  40. Hryniewicz, U., Macarini, L.: Local contact homology and applications, Preprint (2012), arXiv:1202.3122

  41. Kang, J.: Equivariant symplectic homology and multiple closed Reeb orbits, Preprint (2013), arXiv:1305.1245

  42. Katok, A.B.: Ergodic perturbations of degenerate integrable Hamiltonian systems. Izv. Akad. Nauk SSSR Ser. Mat. 37, 539–576 (1973)

    MathSciNet  Google Scholar 

  43. Kerman, E.: On primes and period growth for Hamiltonian diffeomorphisms. J. Mod. Dyn. 6, 41–58 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  44. Kwon, M., van Koert, O.: Brieskorn manifolds in contact and symplectic topology, Preprint (2013). arXiv:1310.0343

  45. Le Calvez, P.: Periodic orbits of Hamiltonian homeomorphisms of surfaces. Duke Math. J. 133, 125–184 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  46. Lerman, E.: Contact toric manifolds. J. Symplectic Geom. 1, 785–828 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  47. Liu, H., Long, Y.: On the existence of two closed characteristics on every compact star-shaped hypersurface in \(\mathbb{R}^4\), Preprint (2013). arXiv:1308.3904

  48. Long, Y.: Index Theory for Symplectic Paths with Applications. Birkäuser, Basel (2002)

    Book  MATH  Google Scholar 

  49. Long, Y.M., Zhu, C.: Ann. Math. (2) 155, 317–368 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  50. Macarini, L., Paternain, G.: Equivariant symplectic homology of Anosov contact structures. Bull. Braz. Math. Soc. (N.S.) 43, 513–527 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  51. Salamon, D., Zehnder, E.: Morse theory for periodic solutions of Hamiltonian systems and the Maslov index. Commun. Pure Appl. Math. 45, 1303–1360 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  52. Taubes, C.H.: The Seiberg–Witten equations and the Weinstein conjecture. Geom. Topol. 11, 2117–2202 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  53. Viterbo, C.: Equivariant Morse theory for starshaped Hamiltonian systems. Trans. Am. Math. Soc. 311, 621–655 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  54. Wang, W.: Existence of closed characteristics on compact convex hypersurfaces in \(\mathbb{R}^{2n}\), Preprint (2011). arXiv:1112.5501

  55. Wang, W., Hu, X., Long, Y.: Resonance identity, stability and multiplicity of closed characteristics on compact surfaces. Duke Math. J. 139, 411–462 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  56. Ziller, W.: Geometry of the Katok examples. Ergod. Theory Dyn. Syst. 3, 135–157 (1983)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

The author is grateful to Alberto Abbondandolo, Viktor Ginzburg, Nancy Hingston and Wolfgang Ziller for useful discussions and to the referee for valuable remarks.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Central Florida, Orlando, FL, 32816, USA

    Başak Z. Gürel

Authors
  1. Başak Z. Gürel
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Başak Z. Gürel.

Additional information

The work is partially supported by the NSF Grant DMS-1414685.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gürel, B.Z. Perfect Reeb flows and action–index relations. Geom Dedicata 174, 105–120 (2015). https://doi.org/10.1007/s10711-014-0006-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10711-014-0006-z

Keywords

Mathematics Subject Classification

Search

Navigation