Symmetric Closed Characteristics on Symmetric Compact Convex Hypersurfaces in $\mathbf{R}^8$

Abstract

Let $\Sigma $ be a $C^3$ compact symmetric convex hypersurface in $\mathbf {R}^{8}$. We prove that when $\Sigma $ carries exactly four geometrically distinct closed characteristics, then all of them must be symmetric. Due to the example of weakly non-resonant ellipsoids, our result is sharp.

Keywords

Compact convex hypersurfaces / Symmetric closed characteristics / Hamiltonian systems / Morse theory / Index iteration theory

References

[1] Ekeland I. Une théorie de Morse pour les systèmes hamiltoniens convexes. Ann. IHP. Anal. non Linéaire.. 1984, 1 19-78

[2] Ekeland I. Convexity Methods in Hamiltonian Mechanics. 1990 Berlin: Springer

[3] Ekeland I, Hofer H. Convex Hamiltonian energy surfaces and their closed trajectories. Commun. Math. Phys.. 1987, 113 419-467

[4] Ekeland I, Lassoued L. Multiplicité des trajectoires fermées d’un systéme hamiltonien sur une hypersurface d’energie convexe. Ann. IHP. Anal. non Linéaire.. 1987, 4 1-29

[5] Fadell E, Rabinowitz P. Generalized cohomological index theories for Lie group actions with an application to bifurcation questions for Hamiltonian systems. Invent. Math.. 1978, 45 2 139-174

[6] Hofer H, Wysocki K, Zehnder E. The dynamics on three-dimensional strictly convex energy surfaces. Ann. Math.. 1998, 148 197-289

[7] Liu C, Long Y, Zhu C. Multiplicity of closed characteristics on symmetric convex hypersurfaces in ${\bf R}^{2n}$. Math. Ann.. 2002, 323 201-215

[8] Long Y. Precise iteration formulae of the Maslov-type index theory and ellipticity of closed characteristics. Adv. Math.. 2000, 154 76-131

[9] Long Y. Index Theory for Symplectic Paths with Applications. Progress in Math. 207. 2002 Basel: Birkhäuser

[10] Long Y, Zhu C. Closed characteristics on compact convex hypersurfaces in ${\bf R}^{2n}$. Ann. Math.. 2002, 155 317-368

[11] Rabinowitz PH. Periodic solutions of Hamiltonian systems. Commun. Pure Appl. Math.. 1978, 31 157-184

[12] Szulkin A. Morse theory and existence of periodic solutions of convex Hamiltonian systems. Bull. Soc. Math. Fr.. 1988, 116 171-197

[13] Wang W. Stability of closed characteristics on compact convex hypersurfaces in ${\bf R}^6$. J. Eur. Math. Soc.. 2009, 11 575-596

[14] Wang W. Symmetric closed characteristics on symmetric compact convex hypersurfaces in ${\bf R}^{2n}$. J. Diff. Equ.. 2009, 246 4322-4331

[15] Wang W. Closed trajectories on symmetric convex Hamiltonian energy surfaces. Discrete Contin. Dyn. Syst.. 2012, 32 2 679-701

[16] Wang, W.: Closed characteristics on compact convex hypersurfaces in ${\bf R}^8$, arXiv:1305.4680v2

[17] Wang W. Irrationally elliptic closed characteristics on compact convex hypersurfaces in ${\bf R}^6$. J. Funct. Anal.. 2014, 267 799-841

[18] Wang W, Hu X, Long Y. Resonance identity, stability and multiplicity of closed characteristics on compact convex hypersurfaces. Duke Math. J.. 2007, 139 3 411-462

[19] Weinstein A. Periodic orbits for convex Hamiltonian systems. Ann. Math.. 1978, 108 507-518