Skip to main content
SpringerLink
Log in

Multiple P-invariant closed characteristics on partially symmetric compact convex hypersurfaces in \({\varvec{R}}^{2n}\)

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

Abstract

In this paper, let \(n\) be a positive integer and \(P=diag(-I_{n-\kappa },I_\kappa ,-I_{n-\kappa },I_\kappa )\) for some integer \(\kappa \in [0, n]\), we prove that for any compact convex hypersurface \(\Sigma \) in \(\mathbf{R}^{2n}\) with \(n\ge 2\) there exist at least two geometrically distinct P-invariant closed characteristics on \(\Sigma \), provided that \(\Sigma \) is P-symmetric, i.e., \(x\in \Sigma \) implies \(Px\in \Sigma \). This work is shown to extend and unify several earlier works on this subject.

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. Borel, A.: Seminar on Transformation Groups. Princeton University Press, Princeton (1960)

    MATH  Google Scholar 

  2. Dong, Y., Long, Y.: Closed characteristics on partially symmetric compact convex hypersurfaces in \({ R}^{2n}\). J. Diff. Eq. 196, 226–248 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  3. Dong, Y., Long, Y.: Stable closed characteristics on partially symmetric compact convex hypersurfaces. J. Diff. Eq. 206, 265–279 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  4. Ekeland, I.: Une théorie de Morse pour les systèmes hamiltoniens convexes. Ann. IHP. Anal. non Linéaire. 1, 19–78 (1984)

    MATH  MathSciNet  Google Scholar 

  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, Berlin (1990)

    Book  MATH  Google Scholar 

  7. Ekeland, I., Hofer, H.: Convex Hamiltonian energy surfaces and their closed trajectories. Comm. 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. Fadell, E., Rabinowitz, P.: Generalized comological index theories for Lie group actions with an application to bifurcation equations for Hamiltonian systems. Invent. Math. 45, 139–174 (1978)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  11. Gromoll, D., Meyer, W.: Periodic geodesics on compact Riemannian manifolds. J. Diff. Geom. 3, 493–510 (1969)

    MATH  MathSciNet  Google Scholar 

  12. Liu, H.: Stability of symmetric closed characteristics on symmetric compact convex hypersurfaces in \({ R}^{2n}\) under a pinching condition. Acta Mathematica Sinica, English Series 28, 885–900 (2012)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  14. Long, Y.: Maslov-type index, degenerate critical points and asymptotically linear Hamiltonian systems. Sci. China. Ser. A. 33, 1409–1419 (1990)

    MATH  MathSciNet  Google Scholar 

  15. Long, Y.: Bott formula of the Maslov-type index theory. Pacific J. Math. 187, 113–149 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  16. 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 

  17. Long, Y.: Index Theory for symplectic paths with applications, progress in Mathemetics, vol. 207. Birkhäuser, Basel (2002)

  18. Long, Y.: Index iteration theory for symplectic paths with applications to nonlinear Hamiltonian systems. In: Proceeedings of International Congress of Mathemetics, vol. II, pp. 303–313. Higher Education Press, Beijing (2002)

  19. Long, Y.: Index iteration theory for symplectic paths and multiple periodic solution orbits. Frontiers Math. China. 1, 178–201 (2006)

    Article  MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  21. Long, Y., Zehnder, E., et al.: Morse theory for forced oscillations of asymptotically linear Hamiltonian systems. In: Albeverio, S. (ed.) Stochastic Proceedings Physics and Geomentry, pp. 528–563. World Scitinfic press, Singapore (1990)

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

  26. Wang, W.: Closed trajectories on symmetric convex Hamiltonian energy surfaces. Discret. Contin. Dyn. Syst. 32(2), 679–701 (2012)

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  28. 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 

Download references

Acknowledgments

I would like to sincerely thank the anonymous referee for his/her careful reading of the manuscript and valuable comments. This paper contains a part of my Ph.D thesis. It is my pleasure to thank my thesis advisor, Professor Yiming Long, for his advices and helps on my study and on this paper.

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui, 230026, People’s Republic of China

    Hui Liu

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

    Hui Liu

Authors
  1. Hui Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hui Liu.

Additional information

Communicated by P. Rabinowitz.

Partially supported by NNSF (No. 11131004), CUSF (No. WK0010000031).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Liu, H. Multiple P-invariant closed characteristics on partially symmetric compact convex hypersurfaces in \({\varvec{R}}^{2n}\) . Calc. Var. 49, 1121–1147 (2014). https://doi.org/10.1007/s00526-013-0614-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00526-013-0614-8

Mathematics Subject Classification (2000)

Search

Navigation