Multiplicity of closed characteristics on symmetric convex hypersurfaces in $\mathbb{R}^{2n}$

Abstract

Let Σ be a compact C 2 hypersurface in R 2n bounding a convex set with non-empty interior. In this paper it is proved that there always exist at least n geometrically distinct closed characteristics on Σ if Σ is symmetric with respect to the origin.

Maslov type index theory for linear Hamiltonian systems with Bolza boundary value conditions and multiple solutions for nonlinear Hamiltonian systems

“…These index theories (about the properties of the index) and their iteration theories have important applications in the study of nonlinear Hamiltonian systems. See [Fei 1995;Conley and Zehnder 1984;Long and Zehnder 1990;Fei and Qiu 1997;Chang et al 1997;Su 1998;Li and Liu 1989] for multiple periodic solutions of asymptotically linear Hamiltonian systems, [Ekeland and Hofer 1985;Dong and Long 1997] for Rabinowitz's minimal periodic problem, and [Ekeland and Hofer 1987;Long and Zhu 2002;Liu et al 2002] for multiple closed characteristics on compact convex hyper-surfaces in ‫ޒ‬ 2n . For a systematic treatment and other applications, one can refer to the excellent books [Ekeland 1990;Long 2002].…”

On the normal exponential map in singular conformal metrics

“…There exists a huge amount of literature concerning the study of brake orbitssee e.g., [12,15,21,22] -and more generally on the study of periodic solutions of autonomous Hamiltonian systems with prescribed energy [13,14,16,17,18]. We also observe here that manifolds with singular boundary of the type investigated in the present paper arise naturally in the study of certain compactifications of incomplete Riemannian manifolds.…”

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

“…Since ∈ SH st (4), by Theorem 1 of [11] and Lemma 4.2 of [19], both (τ 1 , y 1 ) and (τ 2 , y 2 ) must be symmetric. Let ψ j = γ j | [0,τ j /2] , then we have γ j | [0,τ j ] = ψ 2 j and γ j (τ j ) = γ j (τ j /2) 2 .…”