Skip to main content
SpringerLink
Log in

Resonance identities and stability of symmetric closed characteristics on symmetric compact star-shaped hypersurfaces

  • Published:
Calculus of Variations and Partial Differential Equations Aims and scope Submit manuscript

Abstract

So far, it is still unknown whether all the closed characteristics on a symmetric compact star-shaped hypersurface \(\Sigma \) in \(\mathbf{R}^{2n}\) are symmetric. In order to understand behaviors of such orbits, in this paper we establish first two new resonance identities for symmetric closed characteristics on symmetric compact star-shaped hypersurface \(\Sigma \) in \(\mathbf{R}^{2n}\) when there exist only finitely many geometrically distinct symmetric closed characteristics on \(\Sigma \), which extend the identity established by Liu and Long (J Differ Equ 255:2952–2980, 2013) of 2013 for symmetric strictly convex hypersurfaces. Then as an application of these identities and the identities established by Liu et al. (J Funct Anal 166:5598–5638, 2014) for all closed characteristics on the same hypersurface, we prove that if there exist exactly two geometrically distinct closed characteristics on a symmetric compact star-shaped hypersuface in \(\mathbf{R}^4\), then both of them must be elliptic.

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. Bangert, V., Long, Y.: The existence of two closed geodesics on every Finsler 2-sphere. Math. Ann. 346, 335–366 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  2. Berestycki, H., Lasry, J.M., Mancini, G., Ruf, B.: Existence of multiple periodic orbits on starshaped Hamiltonian systems. Commun. Pure Appl. Math. 38, 253–289 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  3. Cristofaro-Gardiner, D., Hutchings, M.: From one Reeb orbit to two. J. Differ. Geom. (to appear) arXiv:1202.4839v2 (2012)

  4. Dell’Antoio, G., D’Onofrio, B., Ekeland, I.: Les systèmes hamiltoniens convexes et pairs ne sont pas ergodiques en général. C. R. Acad. Sci. Paris Sér. I(315), 1413–1415 (1992)

  5. Ekeland, I.: An index theory for periodic solutions of convex Hamiltonian systems. Proc. Symp. Pure Math. 45, 395–423 (1986)

    Article  MathSciNet  Google Scholar 

  6. Ekeland, I.: Convexity Methods in Hamiltonian Mechanics. Springer-Verlag, Berlin (1990)

    Book  MATH  Google Scholar 

  7. Ekeland, I., Hofer, H.: Convex Hamiltonian energy surfaces and their closed trajectories. Commun. Math. Phys. 113, 419–467 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  8. Ekeland, I., Lassoued, L.: Multiplicité des trajectoires fermées d’un systéme hamiltonien sur une hypersurface d’energie convexe. Ann. IHP. Anal. non Linéaire. 4, 1–29 (1987)

    MathSciNet  Google Scholar 

  9. Ginzburg, V.L., Goren, Y.: Iterated index and the mean Euler characteristic. J. Topol. Anal. 7, 453–481 (2015)

    Article  MathSciNet  Google Scholar 

  10. 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 

  11. Girardi, M.: Multiple orbits for Hamiltonian systems on starshaped ernergy surfaces with symmetry. Ann. IHP. Analyse non linéaire. 1, 285–294 (1984)

    MATH  MathSciNet  Google Scholar 

  12. Gromoll, D., Meyer, W.: On differentiable functions with isolated critical points. Topology 8, 361–369 (1969)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  14. Hu, X., Long, Y.: Closed characteristics on non-degenerate star-shaped hypersurfaces in \(\mathbf{R}^{2n}\). Sci. China Ser. A 45, 1038–1052 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  15. Liu, H., Long, Y.: Resonance identity for symmetric closed characteristics on symmetric convex Hamiltonian energy hypersurfaces and its applications. J. Differ. Equ. 255, 2952–2980 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  16. Liu, H., Long, Y.: The existence of two closed characteristics on every compact star-shaped hypersurface in \(\mathbf{R}^4\). Acta Math. Sin. (2014). doi:10.1007/s10114-014-4108-1

  17. Liu, H., Long, Y., Wang, W.: Resonance identities for closed characteristics on compact star-shaped hypersurfaces in \(\mathbf{R}^{2n}\). J. Funct. Anal. 166, 5598–5638 (2014)

  18. Liu, H., Long, Y., Wang, W., Zhang, P.: Symmetric closed characteristics on symmetric compact convex hypersurfaces in \(\mathbf{R}^8\). Commun. Math. Stat. 2, 393–411 (2014)

    Article  MathSciNet  Google Scholar 

  19. Liu, C., Long, Y., Zhu, C.: Multiplicity of closed characteristics on symmetric convex hypersurfaces in \(\mathbf{R}^{2n}\). Math. Ann. 323, 201–215 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  20. Long, Y.: Hyperbolic closed characteristics on compact convex smooth hypersurfaces in \(\mathbf{R}^{2n}\). J. Differ. Equ. 150, 227–249 (1998)

    Article  MATH  Google Scholar 

  21. Long, Y.: Precise iteration formulae of the Maslov-type index theory and ellipticity of closed characteristics. Adv. Math. 154, 76–131 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  22. Long, Y.: Index Theory for Symplectic Paths with Applications. Progress in Math. 207, Birkhäuser, Basel (2002)

  23. Long, Y., Zhu, C.: Closed characteristics on compact convex hypersurfaces in \(\mathbf{R}^{2n}\). Ann. Math. 155, 317–368 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  24. Rabinowitz, P.: Periodic solutions of Hamiltonian systems. Commun. Pure Appl. Math. 31, 157–184 (1978)

    Article  MathSciNet  Google Scholar 

  25. Rademacher, H.-B.: Morse Theorie und geschlossene Geodatische, Bonner Math. Schriften Nr. 229 (1992)

  26. Szulkin, A.: Morse theory and existence of periodic solutions of convex Hamiltonian systems. Bull. Soc. Math. Fr. 116, 171–197 (1988)

    MATH  MathSciNet  Google Scholar 

  27. Viterbo, C.: Une théorie de Morse pour les systèmes hamiltoniens étoilés. C. R. Acad. Sci. Paris Ser. I Math. 301, 487–489 (1985)

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

    Article  MATH  MathSciNet  Google Scholar 

  29. Wang, W.: Stability of closed characteristics on compact convex hypersurfaces in \(\mathbf{R}^6\). J. Eur. Math. Soc. 11, 575–596 (2009)

    Article  MATH  Google Scholar 

  30. Wang, W.: Existence of closed characteristics on compact convex hypersurfaces in \({\bf R}^{2n}\). (2011) arXiv:1112.5501v3

  31. Wang, W.: Closed trajectories on symmetric convex Hamiltonian energy surfaces. Discrete Continuous Dyn. Syst. 32(2), 679–701 (2012)

    Article  MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  33. Weinstein, A.: Periodic orbits for convex Hamiltonian systems. Ann. Math. 108, 507–518 (1978)

    Article  MATH  Google Scholar 

Download references

Acknowledgments

The authors sincerely thank the referee for his/her valuable comments and suggestions on this paper.

Author information

Authors and Affiliations

  1. Key Laboratory of Wu Wen-Tsun Mathematics, Chinese Academy of Sciences, School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, Anhui, People’s Republic of China

    Hui Liu

  2. Chern Institute of Mathematics and LPMC, Nankai University, Tianjin, 300071, People’s Republic of China

    Yiming Long

Authors
  1. Hui Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yiming Long
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yiming Long.

Additional information

Communicated by P. Rabinowitz.

Hui Liu: Partially supported by NSFC (No. 11401555), China Postdoctoral Science Foundation No. 2014T70589, CUSF (No. WK3470000001). Yiming Long: Partially supported by NSFC (No. 11131004), MCME and LPMC of MOE of China, Nankai University and BCMIIS of Capital Normal University.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Liu, H., Long, Y. Resonance identities and stability of symmetric closed characteristics on symmetric compact star-shaped hypersurfaces. Calc. Var. 54, 3753–3787 (2015). https://doi.org/10.1007/s00526-015-0921-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00526-015-0921-3

Keywords

Mathematics Subject Classification

Search

Navigation