## Duke Mathematical Journal

### Resonance identity, stability, and multiplicity of closed characteristics on compact convex hypersurfaces

#### Abstract

There is a long-standing conjecture in Hamiltonian analysis which claims that there exist at least $n$ geometrically distinct closed characteristics on every compact convex hypersurface in ${\bf R}^{2n}$ with $n\ge 2$. Besides many partial results, this conjecture has been completely solved only for $n=2$. In this article, we give a confirmed answer to this conjecture for $n=3$. In order to prove this result, we establish first a new resonance identity for closed characteristics on every compact convex hypersurface ${\Sigma}$ in ${\bf R}^{2n}$ when the number of geometrically distinct closed characteristics on ${\Sigma}$ is finite. Then, using this identity and earlier techniques of the index iteration theory, we prove the mentioned multiplicity result for ${\bf R}^6$. If there are exactly two geometrically distinct closed characteristics on a compact convex hypersuface in ${\bf R}^4$, we prove that both of them must be irrationally elliptic

#### Article information

Source
Duke Math. J., Volume 139, Number 3 (2007), 411-462.

Dates
First available in Project Euclid: 24 August 2007

https://projecteuclid.org/euclid.dmj/1187916266

Digital Object Identifier
doi:10.1215/S0012-7094-07-13931-0

Mathematical Reviews number (MathSciNet)
MR2350849

Zentralblatt MATH identifier
1139.58007

#### Citation

Wang, Wei; Hu, Xijun; Long, Yiming. Resonance identity, stability, and multiplicity of closed characteristics on compact convex hypersurfaces. Duke Math. J. 139 (2007), no. 3, 411--462. doi:10.1215/S0012-7094-07-13931-0. https://projecteuclid.org/euclid.dmj/1187916266

#### References

• V. Bangert and Y. Long, The existence of two closed geodesics on every Finsler 2-sphere, preprint, 2005.
• A. Borel, Seminar on Transformation Groups, with G. Bredon, E. E. Floyd, D. Montgomery, and R. Palais, Ann. of Math. Stud. 46, Princeton Univ. Press, Princeton, 1960.
• G. E. Bredon, Introduction to Compact Transformation Groups, Pure Appl. Math. 46, Academic Press, New York, 1972.
• K.-C. Chang, Infinite-Dimensional Morse Theory and Multiple Solution Problems, Progr. Nonlinear Differential Equations Appl. 6, Birkhäuser, Boston, 1993.
• C. Conley and E. Zehnder, Morse-type index theory for flows and periodic solutions for Hamiltonian equations, Comm. Pure. Appl. Math. 37 (1984), 207--253.
• I. Ekeland, Une théorie de Morse pour les systèmes hamiltoniens convexes, Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), 19--78.
• —, An index theory for periodic solutions of convex Hamiltonian systems'' in Nonlinear Functional Analysis and Its Applications, Part I (Berkeley, Calif., 1983), Proc. Sympos. Pure Math. 45, Part 1, Amer. Math. Soc., Providence, 1986, 395--423.
• —, Convexity Methods in Hamiltonian Mechanics, Ergeb. Math. Grenzgeb. (3) 19, Springer, Berlin. 1990.
• I. Ekeland and H. Hofer, Convex Hamiltonian energy surfaces and their closed trajectories, Comm. Math. Phys. 113 (1987), 419--469.
• I. Ekeland and L. Lassoued, Multiplicité des trajectoires fermées de systéme hamiltoniens convexes, Ann. Inst. H. Poincaré Anal. Non Linéaire 4 (1987), 307--335.
• E. R. Fadell and P. H. Rabinowitz, Generalized comological index theories for Lie group actions with an application to bifurcation equations for Hamiltonian systems, Invent. Math. 45 (1978), 139--174.
• D. Gromoll and W. Meyer, On differentiable functions with isolated critical points, Topology 8 (1969), 361--369.
• —, Periodic geodesics on compact riemannian manifolds, J. Differential Geometry 3 (1969), 493--510.
• J. Han and Y. Long, Normal forms of symplectic matrices, II, Acta Sci. Nat. Univ. Nankai. 32 (1999), 30--41.
• N. Hingston, Equivariant Morse theory and closed geodesics, J. Differential Geom. 19 (1985), 85--116.
• M. W. Hirsch, Differential Topology, Grad. Texts in Math. 33, Springer, New York, 1976.
• H. Hofer, K. Wysocki, and E. Zehnder, The dynamics on three-dimensional strictly convex energy surfaces, Ann. of Math. (2) 148 (1998), 197--289.
• X. Hu and Y. Long, Closed characteristics on non-degenerate star-shaped hypersurfaces in $\Bbb R^2n$, Sci. China Ser. A 45 (2002), 1038--1052.
• W. Klingenberg, Lectures on Closed Geodesics, Grundlehren Math. Wiss. 230, Springer, Berlin, 1978.
• —, Riemannian Geometry, 2nd ed., de Gruyter Stud. Math. 1, de Gruyter, Berlin, 1995.
• Y. M. Long, Maslov-type index, degenerate critical points, and asymptotically linear Hamiltonian systems, Sci. China Ser. A 33 (1990), 1409--1419.
• —, Bott formula of the Maslov-type index theory, Pacific J. Math. 187 (1999), 113--149.
• —, Precise iteration formulae of the Maslov-type index theory and ellipticity of closed characteristics, Adv. Math. 154 (2000), 76--131.
• —, Index iteration theory for symplectic paths with applications to nonlinear Hamiltonian systems'' in Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), Higher Ed. Press, Beijing, 2002, 303--313.
• —, Index Theory for Symplectic Paths with Applications, Progr. Math. 207, Birkhäuser, Basel, 2002.
• —, Index iteration theory for symplectic paths and multiple periodic solution orbits, Front. Math. China 1 (2006), 178--200.
• Y. [Y. M.] Long and D. Dong, Normal forms of symplectic matrices, Acta Math. Sin. (Engl. Ser.) 16 (2000), 237--260.
• Y. M. Long and E. Zehnder, Morse-theory for forced oscillations of asymptotically linear Hamiltonian systems'' in Stochastic Processes, Physics and Geometry (Ascona/Locarno, Switzerland, 1988), World Sci., Teaneck, N.J., 1990, 528--563.
• Y. [Y. M.] Long and C. Zhu, Closed characteristics on compact convex hypersurfaces in $\R^2n$, Ann. of Math. (2) 155 (2002), 317--368.
• J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, Appl. Math. Sci. 74, Springer, New York, 1989.
• P. H. Rabinowitz, Periodic solutions of Hamiltonian systems, Comm. Pure Appl. Math. 31 (1978), 157--184.
• H.-B. Rademacher, On the average indices of closed geodesics, J. Differential Geom. 29 (1989), 65--83.
• —, Morse-Theorie und geschlossene Geodätische, Bonner Math. Schriften 229, Mathematisches Institut, Univ. Bonn, Bonn, 1992.
• E. H. Spanier, Algebraic Topology, McGraw-Hill, New York, 1966.
• A. Szulkin, Morse theory and existence of periodic solutions of convex Hamiltonian systems, Bull. Soc. Math. France 116 (1988), 171--197.
• C. Viterbo, Equivariant Morse theory for starshaped Hamiltonian systems, Trans. Amer. Math. Soc. 311 (1989), 621--655.
• —, A new obstruction to embedding Lagrangian tori, Invent. Math. 100 (1990), 301--320.
• A. Wasserman, Morse theory for G-manifolds, Bull. Amer. Math. Soc. 71 (1965), 384--388.
• A. Weinstein, Periodic orbits for convex Hamiltonian systems, Ann. of Math. (2) 108 (1978), 507--518.